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train | 0.25.3 | Define a l-form on $M$ by
$$\omega(X):=\langlegle d\phi(X),\tau(\phi)\ranglegle,~(X\in TM).$$
Then
\begin{eqnarray*}
\int_M|\omega|dv_g&=&\int_M(\,\,\,\,um_{i=1}^m|\omega(e_i)|^2)^\frac{1}{2}dv_g
\\&\leq&\int_M|\tau(\phi)||d\phi|dv_g
\\&\leq&c Vol(M)^{1-\frac{1}{m}}(\int_M|d\phi|^mdv_g)^\frac{1}{m}
\\&<&\infty.
\end{eqnarray*}
In addition, we calculate $-\delta\omega=\,\,\,\,um_{i=1}^m(\nabla_{e_i}\omega)(e_i)$:
\begin{eqnarray*}
-\delta\omega&=&\,\,\,\,um_{i=1}^m\nabla_{e_i}(\omega(e_i))-\omega(\nabla_{e_i}e_i)
\\&=&\,\,\,\,um_{i=1}^m\{\langlegle\bar{\nabla}_{e_i}d\phi(e_i),\tau(\phi)\ranglegle
-\langlegle d\phi(\nabla_{e_i}e_i),\tau(\phi)\ranglegle\}
\\&=&\,\,\,\,um_{i=1}^m\langlegle \bar{\nabla}_{e_i}d\phi(e_i)-d\phi(\nabla_{e_i}e_i),\tau(\phi)\ranglegle
\\&=&|\tau(\phi)|^2,
\end{eqnarray*}
where in the second equality we used $\bar{\nabla}\tau(\phi)=0$. Therefore $$\int_M|\delta\omega|dv_g=c^2Vol(M)<\infty.$$
Now by Gaffney's theorem and the above equality we have that
$$0=\int_M(-\delta\omega)dv_g=\int_M|\tau(\phi)|^2dv_g=c^2Vol(M),$$
which implies that $c=0$, a contradiction. Therefore we must have $M_1=M$, i.e. $\phi$ is a harmonic map. This completes the proof of Theorem \ref{main1}.
$
\Box$\\
\,\,\,\,ection{Applications to biharmonic submersions}
In this section we give some applications of our result to biharmonic submersions.
First we recall some definitions \cite{BW}.
Assume that $\phi: (M, g)\to (N, h)$ is a smooth map between Riemannian manifolds and $x\in M$. Then $\phi$ is called {\bf horizontally weakly conformal} if either
(i) $d\phi_x=0$, or
(ii) $d\phi_x$ maps the horizontal space $\rm \mathcal{H}_x=\{Ker~d\phi_x\}^\bot$ conformally \textbf{onto} $T_{\phi(x)}N$, i.e.
$$h(d\phi_x(X), d\phi_x(Y))=\lambda^2 g(X, Y), (X, Y\in \mathcal{H}_x),$$
for some $\lambda=\lambda(x)>0,$ called the {\bf dilation} of $\phi$ at $x$.
A map $\phi$ is called {\bf horizontally weakly conformal} or {\bf semiconformal} on $M$ if it is horizontally weakly conformal at every point of $M$. Furthermore, if $\phi$ has no critical points, then we call it a {\bf horizontally conformal submersion}: In this case the dilation $\lambda:M \to (0,\infty)$ is a smooth function. Note that if $\phi: (M, g)\to (N, h)$ is a horizontally weakly conformal map and $\dim M<\dim N$, then $\phi$ is a constant map.
If for every harmonic function $f: V\to \mathbb{R}$ defined on an open subset $V$ of $N$ with $\phi^{-1}(V)$ nonempty, the composition $f\circ\phi$ is harmonic on $\phi^{-1}(V)$, then $\phi$ is called a {\bf harmonic morphism}. Harmonic morphisms are characterized as follows (cf. \cite{Fu, Is}).
\begin{thm}[\cite{Fu, Is}]\label{thm4}
A smooth map $\phi: (M, g)\to (N, h)$ between Riemannian manifolds is a harmonic morphism if and only if $\phi$ is both harmonic and horizontally weakly conformal.
\end{thm}
When $\phi:(M^m, g)\to (N^n, h),(m>n\geq2)$ is a horizontally conformal submersion, the tension field is given by
\begin{eqnarray}\label{eq5}
\tau(\phi)=\frac{n-2}{2}\lambda^2d\phi(grad_\mathcal{H}(\frac{1}{\lambda^2}))
-(m-n)d\phi(\hat{H}),
\end{eqnarray}
where $grad_\mathcal{H}(\frac{1}{\lambda^2})$ is the horizontal component of $\rm grad(\frac{1}{\lambda^2})$, and $\hat{H}$ is the {\bf mean curvature} of the fibres given by the trace
$$\hat{H}=\frac{1}{m-n}\,\,\,\,um_{i=n+1}^m\mathcal{H}(\nabla_{e_i}e_i).$$
Here, $\{e_i, i=1,...,m\}$ is a local orthonormal frame field on $M$ such that $\{e_{i}, i=1,...,n\}$ belongs to $\mathcal{H}_x$ and $\{e_{j}, j=n+1,...,m \}$ belongs to $\mathcal{V}_x$ at each point $x\in M$, where $T_xM=\mathcal{H}_x\oplus \mathcal{V}_x$.
Nakauchi et al. \cite{NUG}, Maeta \cite{Ma} and Luo \cite{Luo2} applied their nonexistence result for biharmonic maps to get conditions for which biharmonic submersions are harmonic morphisms.
Here, we give another such result by using Theorem \ref{main1}. We have
\begin{pro}
Let $\phi:(M^m, g)\to (N^n, h), (m>n\geq2)$ be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold $(M,g)$ with infinite volume,
that admits an Euclidean Sobolev type inequality of the form \eqref{sobolev-inequality},
into a Riemannian manifold $(N, h)$ with sectional curvatures $K^N\leq A$ and $p$ a real constant satisfying $1<p<\infty$. If
$$ \int_M\lambda^p|\frac{n-2}{2}\lambda^2grad_\mathcal{H}(\frac{1}{\lambda^2})
-(m-n)\hat{H}|_g^pdv_g<\infty,$$
and $$\int_M\lambda^mdv_g<\epsilon$$ for sufficiently small $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)), then $\phi$ is a harmonic morphism.
\end{pro}
\proof By (\ref{eq5}) we have,
$$\int_M|\tau(\phi)|_h^pdv_g=\int_M\lambda^p|\frac{n-2}{2}\lambda^2grad_\mathcal{H}(\frac{1}
{\lambda^2})
-(m-n)\hat{H}|_g^pdv_g<\infty,$$
and since $|d\phi(x)|^2=n\lambda^2(x)$, we get that $\phi$ is harmonic by Theorem \ref{main1}. Since $\phi$ is also a horizontally conformal submersion, $\phi$ is a harmonic morphism by Theorem \ref{thm4}. $
\Box$\\
In particular, if $\dim N=2$, we have
\begin{cor}
Let $\phi:(M^m, g)\to (N^2, h)$ be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold $(M, g)$ with infinite volume,
that admits an Euclidean Sobolev type inequality of the form \eqref{sobolev-inequality},
into a Riemannian surface $(N, h)$ with Gauss curvature bounded from above and $p$ a real constant satisfying $1<p<\infty$. If
$$\int_M\lambda^p|\hat{H}|_g^pdv_g<\infty,$$
and $$\int_M\lambda^mdv_g<\epsilon$$ for sufficiently small $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)), then $\phi$ is a harmonic morphism.
\end{cor}
\quad\\
\textbf{Acknowledgements:}
The first named author gratefully acknowledges the support of the Austrian Science Fund (FWF)
through the project P30749-N35 ``Geometric variational problems from string theory''.
The second named author is supported by the NSF of China(No.11501421, No.11771339). Part of the work was finished when the second named author
was a visiting scholar at Tsinghua University. He would like to express his gratitude to
Professor Yuxiang Li and Professor Hui Ma for their invitation and to Tsinghua University for their hospitality. The second named author also would like to thank Professor Ye Lin Ou for his interest in this work and discussion.
{
}
\,\,\,\,c
Volker Branding
Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
{\tt [email protected]}
\,\,\,\,c
Yong Luo
School of mathematics and statistics,
Wuhan university, Wuhan 430072, China
{\tt [email protected]}
\,\,\,\,c
\end{document} | 2,475 | 14,169 | en |
train | 0.26.0 | \begin{document}
\title{Regular Variation in a \rev{Fixed-Point} Problem for Single- and Multiclass Branching Processes and Queues }
\authorone[Aarhus University]{S\O ren Asmussen}
\addressone{Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark}
\authortwo[Heriot-Watt University and\\
. \hspace{3.1cm} Novosibirsk State University]{Sergey Foss}
\addresstwo{School of Mathematical and Computer Sciences, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom. Research supported by RSF grant No. 17-11-01173.}
\begin{abstract}
Tail asymptotics of the solution $R$ to a \rev{fixed-point} problem of type $R \ \mathrm{e}qdistr\ Q + \sum_1^N R_m$
is derived under heavy-tailed conditions allowing both dependence between $Q$ and $N$ and the
tails to be of the same order of magnitude. Similar results are derived for a $K$-class version with applications
to multitype branching processes and busy periods in multiclass queues.\mathrm{e}nd{abstract}
\keywords{Busy period; Galton-Watson process; intermediate regular variation; multivariate regular variation;
random recursion; random sums}
\ams{60H25}{60J80; 60K25; 60F10}
\section{Introduction}\label{S:Intr}
\setcounter{equation}{0}
This paper is concerned with the tail asymptotics of the solution $R$
to the \rev{fixed-point} problem
\begin{equation}\label{eq1}
R \ \mathrm{e}qdistr\ Q + \sum_{m=1}^N R_m
\mathrm{e}nd{equation}
under suitable regular variation (RV) conditions
and the similar problem in a multidimensional setting stated below as \mathrm{e}qref{30.6a}. Here
in \mathrm{e}qref{eq1} $Q,N$ are (possibly dependent) non-negative non-degenerate r.v.'s where $N$ is integer-valued, $R_1,R_2,\ldots$
are i.i.d.\ and distributed as $R$, and ${\overline n}=\mathbb{E} N<1$ (similar notation for expected values is used in the following).
A classical example is $R$ being the M/G/1 busy period, cf.\ \cite{deMeyer}, \cite{Bert}, where $Q$ is the service
time of the first customer in the busy period and $N$ the number of arrivals during his service. Here
$Q$ and $N$ are indeed heavily dependent, with tails of the same order of magnitude when $Q$ has a regularly varying (RV) distribution; more precisely,
$N$ is Poisson$(\lambda q)$ given $Q=q$. Another example is the total progeny of a subcritical branching
process, where $Q\mathrm{e}quiv 1$ and $N$ is the number of children of the ancestor, More generally, $R$ could be
the total life span of the individuals in a Crump-Mode-Jagers process (\cite{Jagers}), corresponding to
$Q$ being the lifetime of the ancestor and $N$ the number of her children. Related examples are weighted
branching processes, see~\cite{Mariana} for references. Note that connections between branching processes and RV
have a long history, going back at least to \cite{Seneta69}, \cite{Seneta74}.
Recall some definitions of classes of heavy-tailed distributions.
A distribution $F$ on the real line is {\it long-tailed}, $F\in {\cal L}$,
if, for some $y>0$
\begin{equation}\label{long}
\frac{\overline{F}(x+y)}{\overline{F}(x)} \rightarrow 1\quad\text{as}\ x\to\infty ;
\mathrm{e}nd{equation}
$F$ is {\it regularly varying}, $F\in {\cal RV}$,
if, for some
$\beta >0$,
$
\overline{F}(x) = x^{-\beta} L(x),
$
where $L(x)$ is a {\it slowly varying} (at infinity) function;\\
$F$ is {\it intermediate regularly varying}, $F\in {\cal IRV}$,
if
\begin{equation}\label{IRV1}
\lim_{\alpha \uparrow 1} \limsup_{x\to\infty}
\frac{\overline{F}(\alpha x)}{
\overline{F}(x)} = 1.
\mathrm{e}nd{equation}
It is known that ${\cal L} \supset {\cal IRV} \supset {\cal RV}$
and if $F$ has a finite mean, then ${\cal L} \supset {\cal S}^*
\supset {\cal IRV}$
where ${\cal S}^*$ is the class of so-called {\it strong subexponential
distributions}, see e.g. \cite{EKM1997} or \cite{FKZ2013} for further
definitions and properties of heavy-tailed distributions.
Tail asymptotics of quantities related to $R$ have earlier been studied in
\cite{Mariana}, \cite{Litvak} under RV conditions (see also~\cite{BDM}).
Our main result is the following:
\begin{theorem}\label{Th:3.8a}
Assume ${\overline n}<1$ and $ {\overline q}<\infty$. Then:\\[1mm]
{\rm (i)}
There is only one \rev{non-negative} solution $R$ to equation \mathrm{e}qref{eq1} with finite mean. For this solution,
${\overline r}={\overline q}/(1-{\overline n})$.\\[1mm]
{\rm (ii)} If further\\
{\bf (C)}
the distribution of $Q+cN$ is intermediate regularly varying for
all $c>0$ in the interval $({\overline r} -\mathrm{e}psilon, {\overline r}+\mathrm{e}psilon )$ where ${\overline r}$ is as in {\rm (i)} and
$\mathrm{e}psilon >0$ is any small number,\\
then
\begin{equation}\label{eq10}
\mathbb{P} (R>x) \sim \frac{1}{1-{\overline n}} \mathbb{P}(Q+{\overline r}
N >x)\quad \text{ as }x\to\infty\,.
\mathrm{e}nd{equation}
{\rm (iii)} In particular, condition {\bf (C)} holds in the following three cases:\\
{\rm (a)} $(Q,N)$ has a 2-dimensional regularly varying distribution;\\
{\rm (b)} $Q$ has an intermediate regularly varying distribution and $\mathbb{P} (N>x) = {\mathrm{o}}(\mathbb{P} (Q>x))$;\\
{\rm (c)} $N$ has an intermediate regularly varying distribution and $\mathbb{P} (Q>x) = {\mathrm{o}}(\mathbb{P} (N>x))$.
\mathrm{e}nd{theorem}
Part (i) is well known from several sources and not deep (see
the proof of the more general
Proposition~\ref{Prop:6.8a} below and the references at the end of
the section for more general versions).
Part (ii) generalizes and unifies results of \cite{Mariana}, \cite{Litvak} in several ways.
Motivated from the Google page rank
algorithm, both of these papers consider a more general recursion
\begin{equation}\label{eq1A}
R \ \mathrm{e}qdistr\ Q + \sum_{m=1}^N A_mR_m\,.
\mathrm{e}nd{equation}
However, \cite{Mariana} does not allow dependence and/or the tails of $Q$ and $N$ to be equally heavy.
These features are incorporated in \cite{Litvak}, but on the other hand that paper require strong conditions
on the $A_i$ which do not allow to take $A_i\mathrm{e}quiv 1$ when dealing with sharp asymptotics. To remove all of these restrictions is essential for
the applications to queues and branching processes we have in mind.
Also, our proofs are considerably simpler and shorter
than those of \cite{Mariana}, \cite{Litvak}. The key tool is a general result of \cite{FossZ}
giving the tail asymptotics of the maximum of a random walk up to a (generalised) stopping time.
\begin{remark}
\rev{In Theorem \ref{Th:3.8a}, we considered the case $A_i\mathrm{e}quiv 1$ only. However, our approach may work in the more
general setting of~\mathrm{e}qref{eq1A} with i.i.d. positive $\{A_m\}$ that do not depend on $Q,N$ and $\{R_m\}$. For example, if we assume, in addition to
${\overline n} <1$,
that $\mathbb{P} (0<A_1\le 1) =1$,
then the exact tail asymptotics for $\mathbb{P} (R>x)$ may be easily found using the upper bound
\mathrm{e}qref{eq10} and the principle of a single big jump. However, the formula
for the tail asymptotics in this case is much more complicated that \mathrm{e}qref{eq10}.}
\mathrm{e}nd{remark}
The multivariate version involves a \rev{set} $\bigl(R(1),\ldots,(R(K)\bigr)$ of r.v.'s satisfying
\begin{equation}\label{30.6a}R(i) \ \mathrm{e}qdistr\ Q(i)+ \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R_{m}(k)\mathrm{e}nd{equation}
In the branching setting, this relates to $K$-type processes by thinking of $N^{(k)}(i)$ as the number of type $k$
children of a type $i$ ancestor.
One example is the total progeny where $Q(i)\mathrm{e}quiv 1$, others relate as above to the total life span and
weighted branching processes.
A queueing example is the busy periods $R(i)$ in the multiclass queue in \cite{AEH}, with $i$ the class of the first
customer in the busy period and $Q(i)$ the service time of a class $i$ customer; the model states that during
service of a class $i$ customer, class $k$ customers arrive at rate $\lambda_{ik}$. One
should note for this example \cite{AEH} \rev{gives} only lower asymptotic bounds,
whereas we here provide sharp asymptotics.
The treatment of \mathrm{e}qref{30.6a} is considerably more involved than for \mathrm{e}qref{eq1}, and we defer the details
of assumptions and results to Section~\ref{S:MRV}. We remark here only that the concept of multivariate
regular variation (MRV) will play a key role; that
the analogue of the crucial assumption ${\overline n}<1$ above is subcriticality, $\rho=$spr$(\mbox{\boldmath$ M$})<1$ where spr means spectral radius and
$\mbox{\boldmath$ M$}$ is the offspring mean matrix with elements
$m_{ik}=\mathbb{E} N^{(k)}(i)$; and that the argument will involve a recursive procedure from \cite{Foss1, Foss2},
reducing $K$ to $K-1$ so that in the end we are back to the case $K=1$ of \mathrm{e}qref{eq1} and Theorem~\ref{Th:3.8a}.
\rev{\subsubsection*{Bibliographical remarks}
An $R$, or its distribution, satisfying \mathrm{e}qref{eq1A} is often called a fixed point of the smoothing transform (going back to~\cite{Liggett}). There is an extensive literature
on this topic, but rather than on tail asymptotics, the emphasis is most often on existence and uniqueness questions
(these are easy in our context with all r.v.'s non-negative with finite mean and we give
short self-contained proofs). Also the assumption $A_i\neq 1$ is crucial for most of this
literature. See further \cite{Aldous}, \cite{Gerold1}, \cite{Gerold2}
and references there.}
\rev{It should be noted that the term ``multivariate smoothing transform"
(e.g.~\cite{Mentemeier}) means to a recursion
of vectors, that is, a version of \mathrm{e}qref{eq1} with $R,Q\in \mathbb{R}^K$. This is different from our set-up because in~\mathrm{e}qref{30.6a} we are only interested in the one-dimensional
distributions of the $R(i)$. In fact, for our applications there is no interpretation of a vector with $i$th marginal having the distribution of $R(i)$.
}
In \cite{Vatutin}, tail asymptotics for the
total progeny of a multitype branching process is studied by
different techniques€‹ in the critical case $\rho =1$. | 3,309 | 22,346 | en |
train | 0.26.1 | \section{One-dimensional case: equation \mathrm{e}qref{eq1}}\label{S:1D}
\setcounter{equation}{0}
The heuristics behind \mathrm{e}qref{eq10} is the principle of a single large jump: for $R$ to exceed $x$, either one or both elements of $(Q,N)$
must be large, or the independent
event occurs that $R_m>x$ for some $m\le N$, in which case $N$ is small or moderate.
If $N$ is large, $\sum_1^NR_m$ is approximately ${\overline r} N$, so roughly the probability of the first possibility is
$\mathbb{P}(Q+{\overline r} N>x)$. On the other hand, results for compound heavy-tailed sums suggest that the approximate probability of the second
possibility is ${\overline n}\mathbb{P}(R>x)$. We thus arrive at
\[\mathbb{P}(R>x) \approx\ \mathbb{P}(Q+{\overline r} N>x)\,+\,{\overline n}\mathbb{P}(R>x)\]
and \mathrm{e}qref{eq10}.
In the proof of Theorem~\ref{Th:3.8a}, let $(Q_1,N_1),(Q_2,N_2),\ldots$ be an i.i.d.\ sequence of pairs distributed as the (possibly dependent)
pair $(Q,N)$ in \mathrm{e}qref{eq1}. Then $S_n=\sum_{i=1}^n \xi_i$, $i=0,1,\ldots$ where $\xi_i=N_i-1$
is a random walk. Clearly,
$\mathbb{E} \xi_i <0$. Let
\begin{equation}\label{eq2}
\tau = \min \{n\ge 1 \ : \ S_n<0\}= \min \{n\ge 1 \ : \ S_n=-1\}\,.
\mathrm{e}nd{equation}
Note that by Wald's identity $\mathbb{E} S_\tau=\mathbb{E}\tau\cdot\mathbb{E} (N-1)$ and $S_\tau=-1$ we have
\begin{equation}\label{3.8b}
\mathbb{E}\tau = \frac{1}{1-\mathbb{E} N}
\mathrm{e}nd{equation}
Now either $N_1=0$, in which case $\tau=1$, or $N_1>0$ so that $S_1=N_1-1$ and to proceed to level -1,
the random walk must go down one level $N_1$ times. This shows that (in obvious notation)
\begin{equation}\label{3.8d}
\tau \ \mathrm{e}qdistr\ 1 + \sum_{i=1}^N \tau_i
\mathrm{e}nd{equation}
That is, $\tau$ is a solution to \mathrm{e}qref{eq1} with
$Q\mathrm{e}quiv 1$. On the other hand, the total progeny in a Galton-Watson process with the number of offsprings
of an individual distributed as $N$ obviously also satisfies \mathrm{e}qref{3.8d}, and hence by uniqueness
must have the same distribution as $\tau$. This result first occurs as equation (4) in \cite{Dwass}, but note that
an alternative representation (1) in that paper appears to have been the one receiving the most attention
in the literature.
Now define $\varphi_i=k_0+k_1Q_i$,
\begin{equation}\label{eq3}
V=\sum_{i=1}^{\tau} \varphi_i
\mathrm{e}nd{equation}
Here the $k_0,k_1$ are non-negative constants, $k_0+k_1>0$.
In particular, if $k_0=1,k_1=0$, then $V=\tau$, and
further
\begin{equation}\label{3.8c}
k_0=0, k_1=1\qquad\Rightarrow\qquad V\mathrm{e}qdistr R.
\mathrm{e}nd{equation}
\rev{Indeed, arguing as before, we conclude that equation $V\mathrm{e}qdistr \varphi +\sum_1^N V_i$ has only one integrable positive solution, and, clearly,
$$
V \mathrm{e}qdistr \varphi + \sum_1^N V_i \mathrm{e}qdistr \varphi + \sum_1^N \varphi_i + \sum_1^N \sum_1^{N_i} \varphi_{i,j} + \sum_1^N \sum_1^{N_i} \sum_1^{N_{i,j}} \varphi_{i,j,k} +
\ldots \mathrm{e}qdistr \sum_1^{\tau} \varphi_i
$$
where, like before, $(\varphi, N)$, $(\varphi_i,N_i)$, $(\varphi_{i,j},N_{i,j})$, etc. are
i.i.d. vectors. In particular, $V$ becomes $R$ when replacing $\varphi$ by $Q$.}
\begin{proof}[Proof of Theorem~\ref{Th:3.8a}]
It remains to find the asymptotics of $\mathbb{P} (V>x)$ as
$x\to\infty$. Throughout the proof, we assume $k_1>0$.
Let $r_0$ be the solution to the
equation
$$
\mathbb{E} \varphi_1+r_0 \mathbb{E} \xi_1=0.
$$
Note that in the particular case where $k_0=0$ and $k_1=1$,
\begin{equation}\label{eq5}
r_0= \frac{\mathbb{E} Q}{1-\mathbb{E} N} = \overline{r}.
\mathrm{e}nd{equation}
Choose $r>r_0$ as close to $r_0$ as needed and let
$$
\psi_i=\varphi_i+ r\xi_i.
$$
We will find upper and lower bounds for the asymptotics
of
$\mathbb{P}(V>x)$ and show that they are asymptotically equivalent.
Since $k_1>0$ and $Q+Nr/k_1$
has an IRV distribution, the distribution of $k_1Q+rN$ is IRV, too.
\\[2mm]
{\bf Upper bound.} The key is to apply the main result of~\cite{FossZ}
to obtain the
following upper bound.
\begin{align*}\MoveEqLeft
\mathbb{P} (V>x) \ = \
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \varphi_i > x\Bigr) \ = \
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \psi_i > x + r
{S}_{\tau}\Bigr)\\
&=\
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \psi_i > x-r\Bigr)\ \le\
\mathbb{P} \Bigl(\max_{1\le k \le \tau}
\sum_{i=1}^{k} \psi_i > x-r\Bigr)\\
&\sim \
\mathbb{E} \tau
\mathbb{P} (\psi_1 > x-r) \ \sim \
\mathbb{E} \tau
\mathbb{P} (\psi_1 > x-r+k_0) \ = \
\mathbb{E} \tau
\mathbb{P} (k_1Q+rN > x)
\mathrm{e}nd{align*}
Here the first equivalence follows from~\cite{FossZ}, noting that the distribution of $\psi_1$
belongs to the class $S^*$ and that~\cite{FossZ} only requires $\varphi_1, \varphi_2,\ldots$ to be i.i.d.\
w.r.t.\ some filtration w.r.t.\ which $\tau$ is a stopping time. For the second, we used
the long-tail property \mathrm{e}qref{long} of the distribution of $\psi_1$.
Let $F$ be the distribution function of $k_1Q+r_0N$.
Then, as $x\to\infty$,
$$
\overline{F}(x) \le \mathbb{P} (k_1Q+rN>x) \le
\mathbb{P} (rk_1Q/r_0+rN>x) \le \overline{F}(\alpha x) \le
(1+o(1)) c(\alpha )\overline{F}(x)
$$
where $\alpha = r_0/r<1$ and $c(\alpha ) =\limsup_{y\to\infty} \overline{F}(\alpha y)/\overline{F}(y)$.
Now we assume the IRV condition to hold, let $r\downarrow r_0$ and apply
\mathrm{e}qref{IRV1} to obtain the upper bound
\begin{equation}\label{eq4}
\mathbb{P} (R>x) \le (1+o(1))
\mathbb{E} \tau \mathbb{P} (k_1Q+r_0N>x)
\mathrm{e}nd{equation}
In particular, if $k_0=0$ and $k_1=1$, then
$r_0=\overline{r}$ is as in~\mathrm{e}qref{eq5}. \\[2mm]
{\bf Lower bound.}
Here we put $\psi_n = \varphi_n + r\xi_n$ where
$r$ is any positive number strictly smaller than $r_0$.
Then the $\psi_n$ are i.i.d.\ random variables with common mean $\mathbb{E} \psi_1 >0$.
We have, for any fixed $C>0$, $L>0$, $n=1,2,\ldots$ and $x\ge 0$ that
\begin{equation}\label{double}
\mathbb{P} (V>x) \ \ge \
\mathbb{P}\Bigl(\sum_{i=1}^{\tau } \psi_i >x\Bigr)\ \ge\ \sum_{i=1}^n \mathbb{P} (D_i \cap A_i)
\mathrm{e}nd{equation}
where
\[
D_i \ =\ \Bigl\{ \sum_{j=1}^{i-1} |\psi_j|\le C, \tau \ge i,
\psi_i>x+C+L \Bigr\}
\quad\text{and}\quad
A_i \ =\ \bigcap_{\mathrm{e}ll\ge 1}\Bigl\{\sum_{j=1}^\mathrm{e}ll \psi_{i+j}\ge -L \Bigr\}.
\]
Indeed, the first inequality in \mathrm{e}qref{double} holds since $S_{\tau}$ is non-positive. Next, the events $D_i$
are disjoint and, given $D_i$, we have $\sum_1^i \psi_j >x+L$. Then, given $D_i\cap A_i$,
we have $\sum_1^k \psi_j \ge x$ for all $k\ge i$ and, in particular, $\sum_{j=1}^{\tau}\psi_j >x$. Thus, \mathrm{e}qref{double} holds.
The events $\{A_i\}$ form a stationary sequence. Due to the SLLN, for any $\varepsilon >0$,
one can choose
$L=L_0$ so large that $\mathbb{P} (A_i) \ge 1-\varepsilon$.
For this $\varepsilon$, choose $n_0$ and $C_0$ such that
$$
\sum_{i=1}^{n_0} \mathbb{P} \Bigl(\sum_{j=1}^{i-1} |\psi_j|\le C_0, \tau \ge i\Bigr)\ \ge\ (1-\varepsilon ) \mathbb{E} \tau.
$$
Since the random variables $(\{\psi_j\}_{j<i},{\mathbf I}(\tau\le i))$ are independent of $\{\psi_j\}_{j\ge i}$, we obtain further that, for any
$\varepsilon \in (0,1)$ and for any $n\ge n_0$, $C\ge C_0$ and $L\ge L_0$,
\begin{eqnarray*}
\mathbb{P} (V>x) &\ge &
\sum_{i=1}^n \mathbb{P} \Bigl(\sum_{j=1}^{i-1} |\psi_j|\le C, \tau \ge i\Bigr)
\mathbb{P} (\psi_i>x+C+L) \mathbb{P} (A_i) \\
&\ge &
(1-\varepsilon )^2 \mathbb{P} (\psi_1 >x+C+L) \sum_{i=1}^n \mathbb{P} (\tau \ge i)\\
&\sim &
(1-\varepsilon )^2 \mathbb{P} (\psi_1 >x) \sum_{i=1}^n \mathbb{P} (\tau \ge i),
\mathrm{e}nd{eqnarray*}
as $x\to\infty$. Here the final equivalence follows from the long-tailedness
of $\psi_1$. Letting first $n$ \rev{go} to infinity and then $\varepsilon$ to zero, we get $\liminf_{x\to\infty} \mathbb{P} (V>x)/\mathbb{E} \tau \mathbb{P} (\psi_1>x) \ge 1.$ Then we let $r\uparrow r_0$ and use the IRV property \mathrm{e}qref{IRV1}. In the particular case
$k_0=0,k_1=1$ we obtain an asymptotic lower bound that is equivalent
to the upper bound derived above
\mathrm{e}nd{proof}
\begin{remark}
A slightly more intuitive approach to the lower bound is to bound $\mathbb{P}(R>x)$ below by the sum of the contributions from the
disjoint events $B_1,B_2,B_3$ where \[
B_1=B\cap\{{\overline r} N\ >\mathrm{e}psilon x\},\quad B_2=B\cap\{A< {\overline r} N\le\mathrm{e}psilon x\},\quad
B_3= \{{\overline r} N\le A\}\]
with $B=\{Q+{\overline r} N>(1+\mathrm{e}psilon) x\}$.
Here for large $x,A$ and small $\mathrm{e}psilon$,
\begin{align*}\mathbb{P}(R>x;\,B_1)\ &\sim\ \mathbb{P}(Q+{\overline r} N>x,\,{\overline r} N>\mathrm{e}psilon x)\\
\mathbb{P}(R>x;\,B_2)\ &\ge \mathbb{P}(Q>x,{\overline r} N\le\mathrm{e}psilon x)\ \sim \ \mathbb{P}(Q+{\overline r} N>x,\,{\overline r} N\le\mathrm{e}psilon x)\\
\mathbb{P}(R>x;B_3)\ &\ge\ \sum_{n=0}^{A/{\overline r}}\mathbb{P}(R_1+\cdots+R_n>x)\mathbb{P}(N=n)\\ & \ \ge\
\sum_{n=0}^{A/{\overline r}}\mathbb{P}\bigl(\max(R_1,\ldots,R_n)>x\bigr)\mathbb{P}(N=n)
\\ & \ \sim\ \sum_{n=0}^{A/{\overline r}}n\mathbb{P}(R>x)\mathbb{P}(N=n)\ \sim\ \mathbb{E}[N\wedge A/{\overline r}]\mathbb{P}(R>x) \ \sim\ {\overline n} \mathbb{P}(R>x)
\mathrm{e}nd{align*}
We omit the full details since they are close to arguments given in Section~\ref{S:FayeMRV} for the
multivariate case.\mathrm{e}nd{remark} | 3,947 | 22,346 | en |
train | 0.26.2 | \section{Multivariate version}\label{S:MRV}
\setcounter{equation}{0}
\rev{The assumptions for \mathrm{e}qref{30.6a} are that all $R_{m}(k)$ are independent of
the vector
\begin{equation}\label{30.6ex}\mbox{\boldmath$ V$}(i)\ =\ \bigl(Q(i),N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)\,,
\mathrm{e}nd{equation}
that they are mutually independent and that $R_{m}(k)\mathrm{e}qdistr R(k)$.
Recall that we are only interested in the one-dimensional
distributions of the $R(i)$. Accordingly, for a solution to \mathrm{e}qref{30.6a}
we only require the validity for each fixed $i$. }
Recall that the offspring mean matrix is denoted $\mbox{\boldmath$ M$}$ where $m_{ik}=\mathbb{E} N^{(k)}(i)$,
and that $\rho=$spr($\mbox{\boldmath$ M$})$; $\rho$ is the Perron-Frobenius root if $\mbox{\boldmath$ M$}$ is irreducible which it is
not necessary to assume. \rev{No restrictions on the dependence structure of the vectors in \mathrm{e}qref{30.6ex} }need to be imposed for the following result to hold (but later we need MRV!):
\begin{proposition}\label{Prop:6.8a} Assume $\rho<1$. Then:\\[1mm]
{\rm (i)} the \rev{fixed-point} problem \mathrm{e}qref{30.6a} has a unique non-negative solution
with ${\overline r}_i=\mathbb{E} R(i)<\infty$ for all $i$;\\[1mm] {\rm (ii)} the ${\overline r}_i=\mathbb{E} R(i)<\infty$ are given as the unique solution to the set
\begin{equation}\label{6.8dd} {\overline r}_i\ =\ {\overline q}_i+\sum_{k=1}^K m_{ik}{\overline r}_k\,,\qquad i=1,\ldots,K,
\mathrm{e}nd{equation}
of linear equations.
\mathrm{e}nd{proposition}
\begin{proof} (i) Assume first $Q(i)\mathrm{e}quiv 1$, $i=1,\ldots,K$. \rev{The existence of a
solution to \mathrm{e}qref{30.6a} is then clear since we may take $R(i)$ as the total progeny
of a type $i$ ancestor in a $K$-type Galton-Watson process where the vector of children of a type $j$ individual is distributed as $\bigl(N^{(1)}(j),\ldots, N^{(K)}(j)\bigr)$.
For uniqueness, let $\bigl(R(1),\ldots, R(K)\bigr)$ be any solution and}
consider the $K$-type Galton-Watson trees ${\mathcal G}(i)$, $ i=1,\ldots,K$, where ${\mathcal G}(i)$ corresponds to an
ancestor of type $i$.
If we define $R^{(0)}(i)=1$,
\[R^{(n)}(i) \ \mathrm{e}qdistr\ 1+ \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R^{(n-1)}_{m}(k)\,,\]
\rev{with similar conventions as for \mathrm{e}qref{30.6a},}
then $R^{(n)}(i)$ is the total progeny of a type $i$ ancestor under the restriction
that the depth of the tree is at most $n$. Induction easily gives that
\rev{$R^{(n)}(i)\preceq_{{\rm st}} R(i)$ ($\preceq_{{\rm st}} \ =\ $ stochastic order) for
each $i$. Since also $R^{(n)}(i) \preceq R^{(n+1)}(i)$, limits
$R^{(\infty)}(i)$ exist, $R^{(\infty)}(i)$ must simply be the unrestricted vector of total progeny of different types,
and $ R^{(\infty)}(i)\preceq_{{\rm st}} R(i)$.
Assuming the $R(i)$} to have finite mean, \mathrm{e}qref{6.8dd} clearly holds with
${\overline q}_i=1$, and so the $\Delta_i={\overline r}_i-\mathbb{E} R^{(\infty)}(i)$ satisfy $\Delta_i=\sum_1^K m_{ik}\Delta_k$. But
$\rho<1$ implies that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible so the only solution is $\Delta_i=0$ which in view of
\rev{$R^{(\infty)}(i)\preceq_{{\rm st}} R(i)$ implies $R^{(\infty)}(i) \mathrm{e}qdistr R(i)$} and the stated uniqueness when $Q(i)\mathrm{e}quiv 1$.
For more general $Q(i)$, we equip each individual of type $j$ in ${\mathcal G}(i)$ with a weight distributed as $Q(j)$,
such that the dependence between her $Q(j)$ and her offspring vector has the given structure. The argument
is then a straightforward generalization and application of what was done above for $Q(i)\mathrm{e}quiv 1$.
(ii) Just take expectations in \mathrm{e}qref{30.6a} and note as before that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible.
\mathrm{e}nd{proof}
For tail asymptotics, we need an MRV assumption. The definition of MRV exists in some equivalent variants,
cf.\ \cite{Sid87}, \cite{Meerschaert}, \cite{Basrak}, \cite{Sid07}, but we shall use the one in polar $L_1$-coordinates
adapted to deal with several random vectors at a time as in \mathrm{e}qref{30.6ex}. Fix here and in the following
a reference RV tail ${\overline F}(x)=L(x)/x^\alpha$ on $(0,\infty)$,
for $\mbox{\boldmath$ v$}=(v_1,\ldots, v_p)$ define $\|\mbox{\boldmath$ v$}\|=\|\mbox{\boldmath$ v$}\|_1=$ $|v_1|+\cdots+|v_p|$ and let ${\cal B}={\cal B}_p=$ $\{\mbox{\boldmath$ v$}:\, \|\mbox{\boldmath$ v$}\|=1\}$.
We then say that a random vector $\mbox{\boldmath$ V$}=(V_1,\ldots, V_p)$
satisfies {\mbox{{\rm MRV}$(F)$}}\ or has property {\mbox{{\rm MRV}$(F)$}}\ if
$\mathbb{P}\bigl(\|\mbox{\boldmath$ V$}\|>x\bigr)$ $\sim b{\overline F}(x)$ where either (1) $b=0$ or (2) $b>0$
and the angular part $\mbox{\boldmath$ T$}heta_{\mbox{\scriptsize\boldmath$ V$}}=\mbox{\boldmath$ V$}/\|\mbox{\boldmath$ V$}\|$ satisfies
\[\mathbb{P}\bigl(\mbox{\boldmath$ T$}heta_{\mbox{\scriptsize\boldmath$ V$}}\in\cdot\,\big|\,\|\mbox{\boldmath$ V$}\|>x\bigr)\ \convdistr \ \mu \text{ as }x\to\infty\]
for some measure $\mu$ on ${\cal B}$ (the angular measure).
Our basic condition is then that for the given reference RV tail ${\overline F}(x)$ we have that\\[1mm]
(MRV) For any $i=1,\ldots,K$ the vector $\mbox{\boldmath$ V$}(i)$ in \mathrm{e}qref{30.6ex} satisfies {\mbox{{\rm MRV}$(F)$}}, where
$b=b(i)>0$ for at least one $i$.\\[1mm]
Note that $F$ is the same for all $i$ but the angular measures $ \mu_i$ not necessarily so.
We also assume that the mean ${\overline z}$ of $F$ is finite, which will ensure that all expected values coming up in the following
are finite.
\rev{Assumption {\mbox{{\rm MRV}$(F)$}}\ implies RV of linear combinations, in particular
marginals. More precisely (see the Appendix),}
\begin{equation}\label{30.6ax}\mathbb{P}\bigl(a_0Q(i)+a_1N^{(1)}(i)+\cdots+a_KN^{(K)}(i) >x\bigr)\ \sim\ c_i(a_0,\ldots,a_K){\overline F}(x)
\mathrm{e}nd{equation} where $\displaystyle
c_i(a_0,\ldots,a_K)\ =\ b(i)\int_{{\mathcal B}}(a_0\theta_0+\cdots+a_K\theta_K)^\alpha\mu_i(\mathrm{d}\theta_0,\ldots,\mathrm{d}\theta_K)\,.$
\begin{theorem}\label{Th:6.8a} Assume that $\rho<1$, ${\overline z}<\infty$ and that \mathrm{e}mph{(MRV)} holds.
Then there are constants $d_1,\ldots,d_K$ such that
\begin{equation}\label{30.6d}
\mathbb{P}(R(i)>x)\,\sim\,d_i{\overline F}(x)\ \ \text{as}\ x\to\infty.
\mathrm{e}nd{equation}
Here the $d_i$ are given as the unique solution to the set
\begin{equation}\label{6.8d} d_i\ =\ c_i(1,{\overline r}_1,\ldots,{\overline r}_K)+\sum_{k=1}^K m_{ik}d_k\,,\qquad i=1,\ldots,K,
\mathrm{e}nd{equation}
of linear equation\rev{s} where the ${\overline r}_i$ are as in Proposition~\ref{Prop:6.8a} and the $c_i$ as in
\mathrm{e}qref{30.6ax}.
\mathrm{e}nd{theorem}
\noindent The proof follows in Sections~\ref{S:Outline}--\ref{S:ProofCompl}. | 2,537 | 22,346 | en |
train | 0.26.3 | \section{Outline of proof}\label{S:Outline}
\setcounter{equation}{0}
When $K>1$, we did not manage to find a random walk argument extending Section~\ref{S:1D}.
Instead, we shall use a recursive procedure, going back to \cite{Foss1, Foss2} in a queueing setting, for eventually
being able to infer \mathrm{e}qref{30.6d}. The identification \mathrm{e}qref{6.8d} of the $d_i$ then follows immediately
from the following result to be proved in Section~\ref{S:FayeMRV} (the case $p=1$ is Lemma 4.7 of \cite{Fay}):
\begin{proposition}\label{Prop:2.8b} Let $\mbox{\boldmath$ N$}=(N_1,\ldots,N_p)$ be MRV with
$\mathbb{P}\bigl(\|\mbox{\boldmath$ N$}\|>x\bigr)\sim c_{\mbox{\scriptsize\boldmath$ N$}}{\overline F}(x)$ and let the r.v.'s $Z_m^{(i)}$, $i=1,\ldots,p$, $m=1,2,\ldots$, be
independent with distribution $F_j$ for $Z_i^{(j)}$, independent of $\mbox{\boldmath$ N$}$
and having finite mean $\overline z_j=\mathbb{E} Z_m^{(j)}$. Define
$S_{m}^{(j)}=Z_1^{(j)}+\cdots+Z_{m}^{(j)}$. If ${\overline F}_j(x)\sim c_j{\overline F}(x)$, then \[
\mathbb{P}\bigl(S_{N_1}^{(1)}+\cdots+S_{N_p}^{(p)}>x\bigr)\ \sim\ \mathbb{P}({\overline z}_1N_1+\cdots+{\overline z}_1N_p\,>\,x)+c_0{\overline F}(x)\]
where $c_0\,=\,c_1\mathbb{E} N_1+\cdots+c_p\mathbb{E} N_p$ . \mathrm{e}nd{proposition}
The recursion idea in \cite{Foss1, Foss2} amounts in a queueing context to let
all class $K$ customers be served first. We implement it here in the branching setting.
Consider the multitype Galton-Watson tree
${\mathcal G}$.
For an ancestors of type $i<K$ and any of her daughters $m=1,\ldots, N^{(K)}(i)$
of type $K$, consider the family tree ${\mathcal G}_m(i)$ formed by $m$ and all her type $K$ descendant in \mathrm{e}mph{direct}
line. For \rev{a vertex} $g\in{\mathcal G}_m(i)$ and $k<K$, let $N^{(k)}_g(K)$ denote the number of type $k$ daughters
of $g$.
Note that ${\mathcal G}_m(i)$ is simply a one-type Galton-Watson tree with the number
of daughters distributed as $N^{(K)}(K)$ and starting from a single ancestor.
In particular, the expected size of ${\mathcal G}_m(i)$ is $1/(1-m_{KK})$. We further have
\begin{align}\label{30.6aa}R(i) \ &\mathrm{e}qdistr\ \widetilde Q(i)+ \sum_{k=1}^{K-1}\sum_{m=1}^{\widetilde N^{(k)}(i)}R_{m}(k)\,,\quad i=1,\ldots,K-1,
\\ \intertext{where} \label{30.6aaa}
\widetilde Q(i)\ &=\ Q(i)+\sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)} Q_g(K)\,, \\ \label{30.6aab}
\widetilde N^{(k)}(i) \ &=\ N^{(k)}(i)+\sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)}N^{(k)}_g(K) \,.
\mathrm{e}nd{align}
that is, a \rev{fixed-point} problem with one type less.
\begin{example}\label{Ex:5.8a}\rm Let $K=2$ and consider the 2-type family tree in Fig.~\ref{treefig},
where type $i=1$ has
green color, the type $2$ descendants of the ancestor in direct line red, and the remaining type 2 individuals blue.
The green type 1 individuals marked with a triangle are the ones that are counted as extra type 1 children
in the reduced recursion \mathrm{e}qref{30.6aa}.
We have $N^{(2)}(1)=2$ and if $m$ is the upper red individual
of type 2, then ${\mathcal G}_m(2)$ has size 4. Further $\sum_{g\in{\mathcal G}_m(1)}N^{(1)}_g=2$, with $m$ herself and her upper
daughter each contributing with one individual.
{\mathcal G}fig
The offspring mean in the reduced 1-type family tree is $\widetilde m=m_{11}+m_{12}m_{21}/(1-m_{22})$.
Indeed, the first term is the expected number of original type 1 offspring of the ancestor and in the second term,
$m_{12}$ is the expected number of type 2 offspring of the ancestor, $1/(1-m_{22})$ the size of the direct line
type 2 family tree produced by each of them, and $m_{21}$ the expected number of type 1 offspring
of each individual in this tree.
Since the original 2-type tree is finite, the reduced 1-type tree must necessarily also
be so, so that $\widetilde m\le 1$. A direct verification of this is instructive. First note that
\[ \widetilde m\le 1\ \iff\ m_{11}-m_{11}m_{22}+m_{12}m_{21}\le 1-m_{22}\ \iff\ \text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\le1\]
But the characteristic polynomial of the
2-type offspring mean matrix $\mbox{\boldmath$ M$}$ is $\lambda^2-\lambda\,\text{tr}(\mbox{\boldmath$ M$})+\text{det}(\mbox{\boldmath$ M$})$.
Further the dominant eigenvalue $\rho$ of $\mbox{\boldmath$ M$}$ satisfies $\rho< 1$ so that
\[ \text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\ \le\ \rho\,\text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\ =\ \rho^2\ <\ 1.\]
\mathrm{e}nd{example} | 1,671 | 22,346 | en |
train | 0.26.4 | \section{Proof of Proposition~\ref{Prop:2.8b}}\label{S:FayeMRV}
\setcounter{equation}{0}
We shall need the following result of Nagaev et al.\ (see the discussion in \cite{Fay}
around equation (4.2) there for references):
\begin{lemma}\label{Lemma:11.7aa} Let $Z_1,Z_2,\ldots$ be i.i.d.\ and RV with finite mean $\overline z$ and
define $S_k=Z_1+\cdots+Z_k$. Then for any $\delta>0$
\[ \sup_{y\ge \delta k}\Bigl|\frac{\mathbb{P}(S_k>k{\overline z}+y)}{k{\overline F}(y)}-1\Bigr|\ \to\ 0,\ k\to\infty.\]
\mathrm{e}nd{lemma}
\begin{corollary}\label{Cor:2.8a} Under the assumptions of Lemma~\ref{Lemma:11.7aa},
it holds for $0<\mathrm{e}psilon<1/{\overline z}$ that
\[d(F,\mathrm{e}psilon)\ =\ \limsup_{x\to\infty}\sup_{k<\mathrm{e}psilon x}\frac{\mathbb{P}(S_{k}>x)}{k{\overline F}(x)}\ < \ \infty\]
\mathrm{e}nd{corollary}
\begin{proof} Define $\delta=(1-\mathrm{e}psilon{\overline z})/\mathrm{e}psilon$. We can write $x=k{\overline z}+y$ where
\[y\ =\ y(x,k)\ =\ x-k{\overline z}\ \ge\ x(1-\mathrm{e}psilon {\overline z})\ =\ x\mathrm{e}psilon\delta\ \ge\ \delta k\,.\]
Lemma~\ref{Lemma:11.7aa} therefore gives that for all large $x$ we can bound $\mathbb{P}(S_{k}>x)$
by $Ck{\overline F}(y)$ where $C$ does not depend on $x$. Now just note that by RV
\[{\overline F}(y) \ \le\ {\overline F}(x\mathrm{e}psilon\delta)\ \sim (\mathrm{e}psilon\delta)^{-\alpha}{\overline F}(x)\,.\]
\mathrm{e}nd{proof}
\begin{proof}[Proof of Proposition~\ref{Prop:2.8b}]
For ease of exposition, we start by the case $p=2$. We split the probability in question into four parts
\begin{align*}p_1(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1\le \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{21}(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{22}(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1\le \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x\bigr)\\
p_3(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x\bigr)
\\ \intertext{Here}
p_1(x)\ &=\ \sum_{k_1,k_2=0}^{\mathrm{e}psilon x}\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)
\mathbb{P}(N_1=k_1,N_2=k_2)
\mathrm{e}nd{align*}
Since $S_{k_1}^{(1)},\,S_{k_2}^{(2)}$ are independent, we have by standard RV theory that
\[\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)\ \sim\ (k_1c_1+k_2c_2){\overline F}(x)\]
as $x\to\infty$. Further Corollary~\ref{Cor:2.8a} gives that for $k_1,k_2\le \mathrm{e}psilon x$ and all large $x$ we have
\begin{align*}\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)\ &\le\
\mathbb{P}\bigl(S_{k_1}^{(1)}>x/2\bigr)+\mathbb{P}\bigl(S_{k_2}^{(2)}>x/2\bigr)\\ &\le\
2\bigl(d(F_1,2\mathrm{e}psilon)k_1+d(F_2,2\mathrm{e}psilon)k_2\bigr){\overline F}(x)\,.
\mathrm{e}nd{align*}
Hence by dominated convergence
\[\frac{p_1(x)}{{\overline F}(x)}\ \to \ \sum_{k_1,k_2=0}^{\infty}(k_1c_1+k_2c_2)
\mathbb{P}(N_1=k_1,N_2=k_2)\ =\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2\,.\]
For $p_3(x)$, denote by $A_j(m)$ the event that $S_{k_j}^{(j)}/k_j\le{\overline z}_j/(1-\mathrm{e}psilon)$
for all $k_j>m$. Then by the LNN there are constants $r(m)$ converging to 0 as $m\to\infty$
such that $\mathbb{P} \rev{\bigl(}A_j(m)^{\mbox{\normalsize c}}\rev{\bigr)} \le r(m)$ for $j=1,2$. It follows that
\begin{align*}
p_3(x)\ &\le \ \bigl(\mathbb{P} \rev{\bigl(}A_1(\mathrm{e}psilon x)^{\mbox{\normalsize c}}\rev{\bigl)}+
\mathbb{P}\rev{\bigl(} A_2(\mathrm{e}psilon x)^{\mbox{\normalsize c}}\rev{\bigl)}\bigr)
\mathbb{P}(N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)\\ &
+\mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x,A_1(\mathrm{e}psilon x),A_2(\mathrm{e}psilon x)\bigr)
\\ &\le r(\mathrm{e}psilon x){\mathrm{O}}\bigl({\overline F}(x)\bigr)+\mathbb{P}\bigl(({\overline z}_1N_1+{\overline z}_2N_2)/(1-\mathrm{e}psilon)>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)\\ &\le\ {\mathrm{o}}\bigl({\overline F}(x)\bigr) \mathbb{P}\bigl(({\overline z}_1N_1+{\overline z}_2N_2)>\mathrm{e}ta x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)
\mathrm{e}nd{align*}
\rev{as $x\to\infty$},
where $\mathrm{e}ta<1-\mathrm{e}psilon$ will be specified later.
For $p_{21}(x)$, we write $p_{21}(x)=p_{21}'(x)+p_{21}''(x)$ where
\begin{align*}
p_{21}'(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,S_{N_2}^{(2)}\le \gamma x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{21}''(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,S_{N_2}^{(2)}> \gamma x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\mathrm{e}nd{align*}
with $\gamma=2\mathrm{e}psilon{\overline z}_2$. Here
\begin{align*}
p_{21}''(x)\ &\le \ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{\mathrm{e}psilon x}^{(2)}>x,S_{\mathrm{e}psilon x}^{(2)}> \gamma x,\,
N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\ &\le\
\mathbb{P}\bigl(S_{ \mathrm{e}psilon x}^{(2)}> \gamma x,\,
N_1> \mathrm{e}psilon x\bigr)\ =\ \mathbb{P}\bigl(S_{ \mathrm{e}psilon x}^{(2)}> \gamma x\Bigr)\,
\mathbb{P}(N_1> \mathrm{e}psilon x)\\ &=\ {\mathrm{o}}(1){\mathrm{O}}\bigl({\overline F}(x)\bigr)\ =\ {\mathrm{o}}\bigl({\overline F}(x)\bigr)\,,
\mathrm{e}nd{align*}
using the LLN in the fourth step. Further as in the estimates above
\begin{align*}
p_{21}'(x)\ &\le\ \mathbb{P}\bigl(S_{N_1}^{(1)}>x(1-\gamma),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\ &\le \
{\mathrm{o}}\bigl({\overline F}(x)\bigr)\ +\ \mathbb{P}\bigl({\overline z}_1N_1>x(1-\gamma)(1-\mathrm{e}psilon),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\\ &\le \ \mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>x(1-\gamma)(1-\mathrm{e}psilon),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\mathrm{e}nd{align*}
We can now finally put the above estimates together. For ease of notation, write $\mathrm{e}ta=\mathrm{e}ta(\mathrm{e}psilon)=
(1-\gamma)(1-\mathrm{e}psilon)$ and note that $\mathrm{e}ta\uparrow 1$ as $\mathrm{e}psilon\downarrow 0$. Using a similar
estimate for $p_{12}(x)$ as for $p_{21}(x)$ and noting that
\[\mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>\mathrm{e}ta x,\, N_1\le \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\ =\ 0\]
for $\mathrm{e}psilon$ small enough, we get
\begin{align*} \MoveEqLeft \limsup_{x\to\infty}\frac{1}{{\overline F}(x)}\mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x\bigr)\\ &
=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2+ \limsup_{x\to\infty}\frac{1}{{\overline F}(x)}\mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>\mathrm{e}ta x\bigr)\\
&=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2 +c({\overline z}_1,{\overline z}_2) \limsup_{x\to\infty}\frac{{\overline F}(\mathrm{e}ta x)}{{\overline F}(x)}\\
&=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2 +c({\overline z}_1,{\overline z}_2)\frac{1}{\mathrm{e}ta^\alpha}
\mathrm{e}nd{align*}
Letting $\mathrm{e}psilon\downarrow 0$ gives that the $\limsup$ is bounded by $c_0\rev{+}c({\overline z}_1,{\overline z}_2)$. Similar estimates
for the $\liminf$ complete the proof for $p=2$.
If $p>2$, the only essential difference is that $p_{21}(x),p_{22}(x)$ need to be replaced by the $2^p-2$ terms
corresponding to all combinations of some $N_i$ being $\le \mathrm{e}psilon x$ and the others $>\mathrm{e}psilon x$,
with the two exceptions being the ones where either all are $\le \mathrm{e}psilon x$ or all are $>\mathrm{e}psilon x$.
However, to each of these similar estimates as the above ones for $p_{21}(x)$ apply.
\mathrm{e}nd{proof} | 3,326 | 22,346 | en |
train | 0.26.5 | \section{Preservation of MRV under sum operations}\label{S:PresMRV}
\setcounter{equation}{0}
Before giving our main auxiliary result, Proposition~\ref{Prop:5.8c}, it is instructive to recall two extremely simple example of MRV. The first is two i.i.d.\ RV$(F)$ r.v.'s $X_1,X_2$, where a big value of the $X_1+X_2$
can only occur if one variable is big and the other small, which gives MRV with the angular measure
concentrated on the points $(1,0),\,(0,1)\in {\cal B}_2$ with mass 1/2 for each. Slightly more complicated:
\begin{proposition}\label{Prop:5.8b}
Let $N,Z,Z_1,Z_2,\ldots$ be non-negative r.v.'s such that $N\in{\mathbb N}$, $Z,Z_1,Z_2,\ldots$ are i.i.d., non-negative
and independent of $N$. Assume that $\mathbb{P}(N>x)\sim c_N{\overline F}(x)$, $\mathbb{P}(Z>x)\sim c_Z{\overline F}(x)$
for some RV \rev{tail ${\overline F}(x)=L(x)/x^\alpha$} and write $S=\sum_1^NZ_i$, ${\overline n}=\mathbb{E} N$, ${\overline z}=\mathbb{E} Z$,
where $c_N+c_Z>0$. Then:\\[1mm]
\mathrm{e}mph{(i)} $\mathbb{P}(S>x)\sim (c_N {\overline z}^{\alpha}+c_Z{\overline n}){\overline F}(x)$;\\[1mm]
\mathrm{e}mph{(ii)} The random vector $(N,S)$ is \mathrm{e}mph{MRV} with \[\mathbb{P}\bigl(\|(N,S)\|>x\bigr)\sim c_{N,S}{\overline F}(x)\quad
\text{where}\quad c_{N,S}=c_N (1+{\overline z}^{\alpha})+c_Z{\overline n}\]
and angular measure $\mu_{N,S}$ concentrated on the points $\mbox{\boldmath$ b$}_1=\bigl(1/(1+{\overline z}),{\overline z}/(1+{\overline z})\bigr)$ and $\mbox{\boldmath$ b$}_2=(0,1)$
with \[\mu_{N,S}(\mbox{\boldmath$ b$}_1)=\frac{c_N}{c_N+c_Z{\overline n}}\,,\qquad \mu_{N,S}(\mbox{\boldmath$ b$}_2)=\frac{c_Z{\overline n}}{c_N+c_Z{\overline n}}\,.\]
\mathrm{e}nd{proposition}
\begin{proof} Part (i) is Lemma 4.7 of~\cite{Fay} (see also~\cite{Denisov}). The proof in~\cite{Fay} also shows that is $S>x$, then
either approximately $N{\overline z}>x$, occuring w.p.\ $c_N{\overline F}(x/{\overline z})\sim c_N {\overline z}^{\alpha}{\overline F}(x)$, or $N\le \mathrm{e}psilon x$ and $Z_i>x$,
occuring w.p.\ $c_Z\mathbb{E}[N\wedge \mathrm{e}psilon x]{\overline F}(x)$. The first possibility is what gives the atom of $\mu_{N,S}$ at $b_1$
and the second gives the atom at $b_2$ since $\mathbb{E}[N\wedge \mathrm{e}psilon x]\uparrow{\overline n}$.
\mathrm{e}nd{proof}
\begin{proposition}\label{Prop:5.8c}
Let $\mbox{\boldmath$ V$}=(\mbox{\boldmath$ T$},N)\in[0,\infty)^{p}\times{\mathbb N}$ satisfy {\mbox{{\rm MRV}$(F)$}},
let $\mbox{\boldmath$ Z$},\mbox{\boldmath$ Z$}_1,\mbox{\boldmath$ Z$}_2,\ldots$\,$\in[0,\infty)^q$ be i.i.d.\ and independent of $(\mbox{\boldmath$ T$},N)$ and
satisfying {\mbox{{\rm MRV}$(F)$}}, and
define $\mbox{\boldmath$ S$}=\sum_1^N\mbox{\boldmath$ Z$}_i$. Then $\mbox{\boldmath$ V$}^*=(\mbox{\boldmath$ T$},N,\mbox{\boldmath$ S$})$ satisfies {\mbox{{\rm MRV}$(F)$}}.
\mathrm{e}nd{proposition}
\begin{proof}
Let $\overline \mbox{\boldmath$ z$}\in[0,\infty)^q$ be the mean of $\mbox{\boldmath$ Z$}$.
Similar arguments as in Section~\ref{S:FayeMRV} show that $\|V^*\|>x$ basically occurs when either
$\|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|>x$ or when $ \|\mbox{\boldmath$ V$}\|\le\mathrm{e}psilon x$ and some $\|\mbox{\boldmath$ Z$}_i\|>x$.
The probabilities of these events are approximately of the form $c'{\overline F}(x)$ and $c''{\overline F}(x)$, so the radial part
of $\mbox{\boldmath$ V$}^*$ is RV with asymptotic tail $c_{\mbox{\scriptsize\boldmath$ V$}^*}{\overline F}(x)$ where $c_{\mbox{\scriptsize\boldmath$ V$}^*}=c'+c''$.
Now
\[\mathbb{P}\Bigl(\frac{(\mbox{\boldmath$ T$},N)}{\|(\mbox{\boldmath$ T$},N)\|}\in \cdot\,\Big|\, \|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|>x\Big)\ \to\ \mu'\]
for some probability measure $\mu'$ on the $(p+1)$-dimensional unit sphere ${\cal B}_{p+1}$; this follows since
$\|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|$ is a norm and the MRV property of a vector is independent of the choice of norm.
Letting $\delta'_0$ be Dirac measure at $(0,\ldots,0)\in\mathbb{R}^q$, $\delta''_0$ be Dirac measure at $(0,\ldots,0)\in\mathbb{R}^{p+1} $
and $\mu''=\mu_{\mbox{\boldmath$ Z$}}$ \rev{the angular measure of $\mbox{\boldmath$ Z$}$}, we obtain the desired conclusion with $c_{\mbox{\scriptsize\boldmath$ V$}^*}=c'+c''$ and \rev{the angular measure of $\mbox{\boldmath$ V$}^*$
given by}
\[\mu_{\mbox{\scriptsize\boldmath$ V$}^*}\ =\ \frac{c'}{c'+c''}\,\mu'\otimes\delta''_0+\frac{c''}{c'+c''}\,\delta'_0\otimes\mu''\]
\mathrm{e}nd{proof}
In calculations to follow (Lemma~\ref{Lemma:7.8a}), extending some $\mbox{\boldmath$ V$}$ to some $\mbox{\boldmath$ V$}^*$ in a number of steps, expressions for $c_{\mbox{\scriptsize\boldmath$ V$}^*},\mu_{\mbox{\scriptsize\boldmath$ V$}^*}$ can be deduced along the lines of the proof of
Propositions~\ref{Prop:5.8b}--\ref{Prop:5.8c} but the expression and details become extremely tedious. Fortunately, they won't be needed and are therefore omitted --- all that matters is existence.
If $\alpha$ is not an even integer,
the MRV alone of $\mbox{\boldmath$ V$}^*$ can alternatively (and slightly easier) be obtained from
Theorem 1.1(iv) of~\cite{Basrak}, stating that by non-negativity it suffices to verify MRV of any linear combination. | 1,941 | 22,346 | en |
train | 0.26.6 | \section{Proof of Theorem~\ref{Th:6.8a} completed}\label{S:ProofCompl}
\setcounter{equation}{0}
\begin{lemma}\label{Lemma:7.8a} In the setting of \mathrm{e}qref{30.6aa},
the random vector \[\mbox{\boldmath$ V$}^*(i)\ =\ \bigl(\widetilde Q(i), \widetilde N^{(1)}(i),\ldots,\widetilde N^{(K-1)}(i)\bigr)\] satisfies {\mbox{{\rm MRV}$(F)$}}\
for all $i$.
\mathrm{e}nd{lemma}
\begin{proof}
Let $\bigl|{\mathcal G}_m(i)\bigr|$ be the
number of elements of ${\mathcal G}_m(i)$ and
\begin{align*} M_1(i)\ &=\ \sum_{m=1}^{N^{(K)}(i)}\bigl|{\mathcal G}_m(i)\bigr|\,,\\
M_2(i)\ &=\ \sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)}\bigl(Q_g(K),N_g^{(1)}(K),\ldots,N_g^{(1)}(K-1)\bigr)\mathrm{e}nd{align*}
Recall that our basic assumption is that the
\begin{equation}\label{7.8d} \mbox{\boldmath$ V$}^*(i)\ =\ \bigl(Q(i), N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)\mathrm{e}nd{equation}
satisfy {\mbox{{\rm MRV}$(F)$}}. The connection to a Galton-Watson tree
and Theorem~\ref{Th:3.8a} with $Q\mathrm{e}quiv 1$, $N=N^{(K)}(i)$ therefore imply that so does any $\bigl|{\mathcal G}_m(i)\bigr|$,
and since these r.v.'s are i.i.d.\ and independent of $N^{(K)}(i)$, Proposition~\ref{Prop:5.8c} gives that
$\mbox{\boldmath$ V$}_1(i)=\bigl(\mbox{\boldmath$ V$}(i), M_1(i)\bigr)$ satisfies {\mbox{{\rm MRV}$(F)$}}. Now the {\mbox{{\rm MRV}$(F)$}}\ property of \mathrm{e}qref{7.8d} with $i=K$
implies that the vectors $\bigl(Q_g(K),N_g^{(1)}(K),\ldots,N_g^{(K-1)}(K)\bigr)$, being distributed as
$\bigl(Q(K),N^{(1)}(K),\ldots,N^{(K-1)}(K)\bigr)$
again satisfy {\mbox{{\rm MRV}$(F)$}}. But $M_2(i)$ is a sum of $M_1(i)$ such vectors that are i.i.d.\ given $M_1(i)$.
Using Proposition~\ref{Prop:5.8c} once more gives that $\mbox{\boldmath$ V$}_2(i)=\bigl(\mbox{\boldmath$ V$}(i), M_1(i),M_2(i)\bigr)$ satisfies {\mbox{{\rm MRV}$(F)$}}.
But $\mbox{\boldmath$ V$}^*(i)$ is a function of $\mbox{\boldmath$ V$}_2(i)$. Since this function is linear, property {\mbox{{\rm MRV}$(F)$}}\ of
$\mbox{\boldmath$ V$}_2(i)$ carries over to $\mbox{\boldmath$ V$}^*(i)$.
\mathrm{e}nd{proof}
\begin{proof}[Proof of Theorem~\ref{Th:6.8a}]
We use induction in $K$. The case $K=1$ is just Theorem~\ref{Th:3.8a}, so assume
Theorem~\ref{Th:6.8a} shown for $K-1$.
The induction hypothesis and Lemma~\ref{Lemma:7.8a} implies that $\mathbb{P}(R(i)>x)\sim d_i{\overline F}(x)$ for $i=1,...,K-1$.
Rewriting \mathrm{e}qref{30.6a} for $i=K$ as
\begin{equation*} R(K) \ \mathrm{e}qdistr\ Q^*(K)+\sum_{m=1}^{N^{(K)}(K)}R_{m}(K)
\ \ \text{where}\ \ Q^*(K)\,=\,\sum_{k=1}^{K-1}\sum_{m=1}^{N^{(k)}(K)}R_{m}(k)\,,\mathrm{e}nd{equation*}
we have a \rev{fixed-point} problem of type \mathrm{e}qref{eq1} and can then use Theorem~\ref{Th:6.8a} to conclude that
also $\mathbb{P}(R(K)>x)\sim d_K{\overline F}(x)$, noting that the needed MRV condition on $\bigl(Q^*(K),N^{(k)}(K)\bigr)$
follows by another application of Proposition~\ref{Prop:5.8c}.
Finally, to identify the $d_i$ via \mathrm{e}qref{6.8d}, appeal to Proposition~\ref{Prop:2.8b} with
$\mbox{\boldmath$ N$}=\bigl(Q(i), N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)$, writing the r.h.s.\ of \mathrm{e}qref{30.6a} as
\[{\mathrm{O}}(1)\,+\,\sum_{m=1}^{\lfloor Q(i)\rfloor} 1\,+\, \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R_{m}(k)\,.\]
Existence and uniqueness of a solution to \mathrm{e}qref{6.8d} follows by once more noticing that $\rho<1$
implies that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible.
\mathrm{e}nd{proof}
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\appendix | 3,654 | 22,346 | en |
train | 0.26.7 | \section{\rev{Proof of \mathrm{e}qref{30.6ax}}}
\rev{The RV of linear combinations subject to MRV assumptions has received considerable attention, see e.g.~\cite{Basrak}, but we could not find explicit
formulas like \mathrm{e}qref{30.6ax} for the relevant constants so we give a
self-contained proof. The formula is a special case of the following: if $\mbox{\boldmath$X$}=(X_1\,\ldots\,X_n)\in\mathbb{R}^n$
is a random vector such that $\mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>t\big)\sim L(t)/t^\alpha$
and $\mbox{\boldmath$ T$}heta=\mbox{\boldmath$X$}/\|\mbox{\boldmath$X$}\|$ has conditional limit distribution
$\mu$ in ${\mathcal B}_1$ given $\|\mbox{\boldmath$X$}\|>t$ as $t\to\infty$, then
\[\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>x)\ =\ \mathbb{P}(a_1X_1+\cdots+a_nX_n>t)\ \sim\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha\mu(\mathrm{d}\mbox{\boldmath$ \theta$})\]}
\rev{To see this, note that given $\mbox{\boldmath$ T$}heta=\mbox{\boldmath$ \theta$}\in{\mathcal B}_1$,
$\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$} =$ $ \|\mbox{\boldmath$X$}\|(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})$
will exceed $t>0$ precisely when $\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0$ and $\|\mbox{\boldmath$X$}\|>t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}$.
Thus one expects that
\begin{align*}\MoveEqLeft
\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)\ \sim\
\int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)\mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\bigr)\,\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\\ & \sim\
\int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)\frac{L(t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})}{(t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha}\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\ \sim\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\mathrm{e}nd{align*}
which is the same as asserted.}
\rev{For the rigorous proof, assume $\|\mbox{\boldmath$a$}\|=1$. Then ${\mathcal B}_1$ is the disjoint union of the sets
$B_{1,n},\ldots,B_{n,n}$ where $B_{i,n}=$
$\{\mbox{\boldmath$ \theta$}\in{\mathcal B}_1:\,(i-1)/n<\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\le i/n\}$
for $i=2,\ldots,n$ and $B_{1,n}=$ $\{\mbox{\boldmath$ \theta$}\in{\mathcal B}_1:\,\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\le 1/n\}$. Assuming $\mathbb{P}(\mbox{\boldmath$ T$}heta=i/n)=0$ for all integers $i,n$, we get
\begin{align*}
\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)\ &=\ \sum_{i=1}^n \mathbb{P}\bigl(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t,\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)\
\le\ \sum_{i=1}^n \mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>ti/n,\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)
\\& \sim\ \sum_{i=1}^n \frac{L(ti/n)}{(ti/n)^\alpha}
\mathbb{P}\bigl(\mbox{\boldmath$ T$}heta\in B_{i,n}\,\big|\,|\mbox{\boldmath$X$}\|>t\bigr)\
\sim\ \frac{L(t)}{t^\alpha} \sum_{i=1}^n (i/n)^\alpha \mathbb{P}\bigl(\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)\\ &=\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}f_{+,n}(\mbox{\boldmath$ \theta$})\,\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\mathrm{e}nd{align*}
where $ f_{+,n}$ is the step function taking value $(i/n)^\alpha$ on $B_{i,n}$.
A similar argument gives the asymptotic lower bound $\int \!f_{-,n}\,\mathrm{d}\mu\,L(t)/t^\alpha$ for
$\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)$ where $ f_{-,n}$ equals $\bigl((i-1)/n)\bigr)^\alpha$ on $B_{i,n}$
for $i>1$ and $0$ on $B_{1,n}$.
But $f_{\pm,n}(\mbox{\boldmath$ \theta$})$ both have limits $\bigl[(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^+\bigr]^\alpha$
as $n\to\infty$ and are
bounded by 1. Letting $n\to\infty$ and using dominated convergence completes the proof.}
\rev{The case $\mathbb{P}(\mbox{\boldmath$ T$}heta=i/n)>0$ for some $i,n$ is handled by a trivial
redefinition of the $B_{i,n}$.}
\mathrm{e}nd{document}
The referee has asked us for a reference for the the value of the
constant $c(a_0,\ldots,a_K)$ in the following result:
if the non-negative vector $(X_0,\ldots,X_K)$ is MRV with index $\alpha$ and angular measure $\mu$, then
\begin{equation}\label{30.6ax}\mathbb{P}\bigl(a_0X_0+a_1X_1+\cdots+a_KX_K >x\bigr)\ \sim\ c(a_0,\ldots,a_K){\overline F}(x)
\mathrm{e}nd{equation} where $\displaystyle
c(a_0,\ldots,a_K)\ =\ (a_0\theta_0+\cdots+a_K\theta_K)^\alpha\mu(\mathrm{d}\theta_0,\ldots,\mathrm{d}\theta_K)\,.$
I have it in my notes used for my course \mathrm{e}mph{Risk}, but when I wrote them,
I didn't see the formula anywhere in the papers on MRV and linear combinations.
It must of course be known.
\mathrm{e}nd{document} | 1,961 | 22,346 | en |
train | 0.27.0 | \begin{document}
\title[Carath\'eodory functions]{Carath\'eodory
functions in the Banach space setting}
\author[D. Alpay]{Daniel Alpay}
\address{Department of Mathematics\\
Ben--Gurion University of the Negev\\
Beer-Sheva 84105\\ Israel} \email{[email protected]}
\thanks{D. Alpay thanks the Earl Katz family for endowing the chair which
supports his research}
\author[O. Timoshenko]{Olga Timoshenko}
\address{Department of Mathematics\\
Ben--Gurion University of the Negev\\
Beer-Sheva 84105\\ Israel} \email{[email protected]}
\author[D. Volok]{Dan Volok}
\address{Department of Mathematics\\
Kansas State University\\
Manhattan, KS 66506-2602\\USA}\email{[email protected]}
\subjclass[2000]{30A86; 47A56.}
\date{}
\begin{abstract}
We prove representation theorems for Carath\'eodory functions in the setting of Banach spaces.
\end{abstract}
\maketitle
\section{Introduction}
L. de Branges and J. Rovnyak introduced in \cite{dbr1}, \cite{dbr2}
various families of reproducing kernel Hilbert spaces of functions which take
values in a Hilbert space and are analytic in the open unit disk
or in the open upper half-plane. These spaces play an important role
in operator theory, interpolation theory,
inverse scattering, the theory of wide sense stationary stochastic
processes and related topics; see for instance \cite{Dmk},
\cite{ad1}, \cite{ad2}. In the case of the open unit disk
$\mathbb{D}$, of particular importance are the following two
kinds of reproducing kernels:
\begin{align}
\label{avron} k_\phi(z,w)&=\frac{\phi(z)+\phi(w)^*}{2(1-zw^*)},\\
\nonumber
k_s(z,w)&=\frac{I-s(z)s(w)^*}{1-zw^*}.
\end{align}
In these expressions, $s(z)$ and $\phi(z)$ are operator-valued functions analytic
in $\mathbb{D}$, $*$ denotes
the Hilbert space adjoint, and $I$ denotes the identity operator.
The functions for which the kernels $k_\phi(z,w)$ and $k_s(z,w)$ are positive
are called respectively Carath\'eodory and Schur functions.
We remark that one can use the Cayley
transform
$$s(z)=(I-\phi(z))(I+\phi(z))^{-1}$$
to reduce the study of the kernels $k_\phi(z,w)$ to the study of
the kernels $k_s(z,w)$. For these latter it is
well known that the positivity of the kernel $k_s(z,w)$ implies analyticity of $s(z)$.\\
Every Carath\'eodory function
admits two equivalent representations. The first, called the
Riesz -- Herglotz representation, reads as follows:
\begin{equation}
\label{anaelle}
\phi(z)=ia+\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}d\mu(t)
\end{equation}
where $a$ is a real number and where $\mu(t)$ is an increasing
function such that $\mu(2\pi)<\infty$. The integral is a
Stieltjes integral and the proof relies on Helly's theorem; see
\cite[pp. 19--27]{MR48:904}.\\
The second representation reads:
\begin{equation}
\label{anaelle2}
\phi(z)=ia+\Gamma(U+zI)(U-zI)^{-1}\Gamma^*
\end{equation}
where $a\in{\mathbb R}$ and where $U$ is a unitary operator in an
auxiliary Hilbert space ${\mathcal H}$ and $\Gamma$ is a bounded
operator from ${\mathcal H}$ into ${\mathbb C}$.\\
The expression \eqref{anaelle2} still makes sense in a more
general setting when the kernel $k_\phi(z,w)$ has a finite
number of negative squares. The space ${\mathcal H}$ is then a
Pontryagin space. This is the setting in which Kre\u\i n and
Langer proved this result; see \cite[Satz 2.2 p. 361]{kl1}. They
allowed the values of the function $\phi(z)$ to be operators
between Pontryagin spaces and required weak continuity at the
origin. Without this hypothesis one can find functions for which
the kernel $k_\phi(z,w)$ has a finite number of negative squares
but which are not meromorphic in ${\mathbb D}$ and in particular
cannot admit representations of the form \eqref{anaelle2}; for
instance the function
$$\phi(z)=\begin{cases} 0\,\,{\rm if}\,\, z\not=0\\
1\,\,{\rm if}\,\, z=0
\end{cases}
$$
defines a kernel $k_\phi(z,w)$ which has one negative square; see
\cite[p. 82]{adrs} for an analogue for $k_s(z,w)$
kernels.\\
Operator--valued Carath\'eodory functions were extensively studied in the
Hilbert space case; see e.g. \cite{MR1821917}, \cite{MR1759548},
\cite{MR899274}, \cite{MR1036844}. We would also like to mention the
non-stationary setting, where an analogue of the representation
\eqref{anaelle2} was obtained for upper-triangular operators; see
\cite{MR2003f:47021} and \cite{MR1704663}.\\
The notion of reproducing kernel space (with positive or
indefinite metric) can also be introduced for functions which
take values in Banach spaces and even topological vector spaces.
The positive case was studied already by Pedrick for functions
with values in certain topological vector spaces in an
unpublished report \cite{pedrick} and studied further by P. Masani in his 1978 paper
\cite{masani78}. Motivations originate from the theory of partial
differential equations (see e.g. \cite{MR1973084}) and the theory
of stochastic processes (see e.g. \cite[\S 4]{MR647140} and \cite{MR626346}).\\
The present paper is devoted to the study of Carath\'eodory functions
whose values are bounded operators between appropriate Banach
spaces. It seems that there are no natural analogs of
Schur functions or of the Cayley transform in this setting.\\
Let ${\mathcal B}$ be a Banach space. We denote by ${\mathcal B}^*$ the
space of anti-linear bounded
functionals (that is, its conjugate dual space). The duality between ${\mathcal B}$
and ${\mathcal B}^*$ is
denoted by
\[
\langle b_*, b\rangle_{\mathcal B}\defi b_*(b),\quad \text{where
} b\in {\mathcal B}\text{ and } b_*\in{\mathcal B}^*.\] An
${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function $\phi(z)$ defined in
some open neighborhood $\Omega$ of the origin
and weakly continuous at the origin will be called a Carath\'eodory
function if the ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
kernel
\begin{equation}
\label{toto}
k_\phi(z,w)=\frac{\phi(z)+\phi(w)^*\big|_{\mathcal B}}{2(1-zw^*)}
\end{equation}
is positive in $\Omega$. The notion of positivity for bounded
operators and kernels from ${\mathcal B}$
into ${\mathcal
B}^*$ is reviewed in the next section. We shall prove (see Theorem \ref{Ivry-sur-Seine})
that every Carath\'eodory function admits a representation of the form
\eqref{anaelle2} and, in particular, admits an analytic extension to
$\mathbb{D}$; see e.g. \cite[pp. 189--190]{MR58:12429a} for
information on vector-valued analytic functions. We note that the proof of
this theorem can be adapted to the case when the kernel
$k_\phi(z,w)$ has a finite number of negative squares;
see Remark \ref{indef}.\\
Furthermore, if $\mathcal{B}$ is a separable Banach space
then ${\mathbf L}({\mathcal B},{\mathcal
B}^*)$-valued Carath\'eodory functions can be characterized as
functions analytic in $\mathbb{D}$ and such that
\[
\phi(z)+\phi(z)^*\bigm|_{\mathcal B}\ge 0,\quad z\in{\mathbb D}.
\]
Moreover, in this case we have an analogue of the Riesz -- Herglotz
representation \eqref{anaelle}; see Theorem
\ref{Antony}.\\
We conclude with the outline of the paper; the next three sections
are of preliminary nature, and deal with positive operators, Stieltjes
integrals and Helly's theorem respectively. Representation theorems for
Carath\'eodory functions are proved in Section 5. Two cases are to be
distinguished, as whether ${\mathcal B}$ is separable or not.
The case of ${\mathbf L}({\mathcal B}^*,{\mathcal B})$-valued
Carath\'eodory functions will be treated in the last section of this paper.
This case is of special importance. Indeed, if $\phi(z)$ is a ${\mathbf L}
({\mathcal B}^*, {\mathcal B})$-valued Carath\'eodory function which takes
invertible values, its inverse is a
${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued Carath\'eodory
function. In the Hilbert space case, this fact has important connections
with operator models for pairs of unitary
operators (see \cite{dbs} and, for the
analogue for self--adjoint operators, \cite{dbr1}, \cite{MR1960423}).
We will explore the Banach space generalizations of these results
in a future publication. | 2,618 | 13,058 | en |
train | 0.27.1 | \section{Positive operators and positive kernels}
In this section we review for the
convenience of the reader various facts on bounded positive operators from
${\mathcal B}$ into ${\mathcal B}^*$.
First some notations and a definition.
\begin{definition} Let ${\mathcal B}$ be a complex Banach space and let
$A\in{\mathcal L}({\mathcal B},{\mathcal B}^*).$ The operator $A$ is said to
be {\em positive} if
\[
\langle Ab,b\rangle_{\mathcal B}\geq 0,\quad \forall b\in {\mathcal B}.\]
\end{definition}
Note that a positive operator is in particular self-adjoint in the sense
that $A=A^*\bigm|_{\mathcal B}$, that
is,
\begin{equation}
\label{gabriel} \langle Ab,c\rangle_{{\mathcal B}}=
\overline{\langle Ac,b\rangle_{{\mathcal B}}}.
\end{equation}
Indeed, \eqref{gabriel} holds for $b=c$ in view of the positivity. It then
holds for all choices of $b,c\in{\mathcal B}$ by polarization:
\[
\begin{split}
\langle Ab,c\rangle_{{\mathcal B}}&=\frac{1}{4} \sum_{k=0}^3i^k\langle A(b+i^kc),(b+i^kc)
\rangle_{{\mathcal B}}\\
&=\frac{1}{4}
\sum_{k=0}^3i^k\langle A(c+i^{-k}b),(c+i^{-k}b)
\rangle_{{\mathcal B}}\\
&=\frac{1}{4}\overline{
\sum_{k=0}^3i^{-k}\langle A(c+i^{-k}b),(c+i^{-k}b)
\rangle_{{\mathcal B}}}\\
&=\overline{\langle Ac,b\rangle_{{\mathcal B}}}.
\end{split}
\]
Now, let $\tau$ be the natural injection
from ${\mathcal B}$ into
${\mathcal B}^{**}$:
\begin{equation}
\label{bastille}
\langle \tau(b), b_*\rangle_{{\mathcal B}^*} =
\overline{\langle b_*, b\rangle_{\mathcal B}}.
\end{equation}
We have for $b,c\in{\mathcal B}$:
\[
\begin{split}
\langle A^*\tau c,b\rangle_{\mathcal B}&=\langle \tau c, Ab
\rangle_{{\mathcal B}^*}\\
&=\overline{\langle Ab,c\rangle_{\mathcal B}}\\
&=\langle Ac,b\rangle_{\mathcal B}
\end{split}
\]
in view of \eqref{gabriel}, and hence $A=A^*\big|_{\mathcal B}$.\\
The following factorization result is well known and originates
with the works of Pedrick \cite{pedrick} (in the case of
topological vector spaces with appropriate properties) and
Vakhania \cite[\S 4.3.2 p.101]{MR626346} (for positive elements
in ${\mathbf L}({\mathcal B}^*,{\mathcal B})$); see the
discussion in \cite[p. 416]{masani78}. We refer also to
\cite{MR647140} for the case of barreled spaces and to
\cite{MR1973084} for the case of unbounded operators.
\begin{theorem}
The operator $A\in{\mathcal L}({\mathcal B},{\mathcal B}^*)$ is
positive if and only if there exist a Hilbert space ${\mathcal H}$
and a bounded operator $T\in{\mathbf L} ({\mathcal B},{\mathcal
H})$ such that $A=T^*T$. Moreover,
\begin{equation}
\label{samantha}
\langle Ab,c\rangle_{{\mathcal B}}=\langle Tb,
Tc \rangle_{\mathcal H},\quad b,c\in {\mathcal B}
\end{equation}
and
\begin{equation}
\label{bagdad} \sup_{\|b\|=1}\langle Ab,b\rangle_{{\mathcal B}} =\|A\|=\|T\|^2.
\end{equation}
Finally we have
\begin{equation}
\label{sof-sof}
|\langle Ab,c\rangle_{\mathcal B}|\le\langle Ab,b\rangle^{1/2}
\langle Ac,c\rangle^{1/2}.
\end{equation}
\label{rosh-hashana}
\end{theorem}
\begin{proof} For $Ab$ and $Ac$ in the range of $A$ the expression
\begin{equation}
\label{mind your step} \langle Ab,Ac\rangle_A= \langle Ab,c\rangle_{{\mathcal B}}=
\overline{\langle
Ac,b\rangle_{{\mathcal B}}}.
\end{equation}
is well defined in the sense that $Ab=0$ (resp. $Ac=0$) implies that
\eqref{mind your step} is equal to $0$.
Thus formula \eqref{mind your step} defines a
sesquilinear form on $\ran A$. It is
positive since the operator $A$ is positive. Moreover, it is non-degenerate because
if $\langle Ab,b\rangle_{\mathcal{B}}=0$ then $Ab=0.$ \\
Indeed, if $c$ is such that $\langle Ac,c\rangle_{\mathcal{B}}=0$ then
\eqref{gabriel} implies that
the real and the imaginary parts of $\langle Ab,c\rangle_{\mathcal{B}}$ are equal, respectively, to
\[\dfrac{1}{2}\langle A(b+c),b+c\rangle_{\mathcal{B}}\quad\text{and}\quad\dfrac{1}{2}\langle A(b+ic),b+ic\rangle_{\mathcal{B}}\] and, therefore, are non-negative. Then the same can be said about
$\langle Ab,-c\rangle_{\mathcal{B}},$ hence
\begin{equation}\label{referee}
\langle Ab,c\rangle_{\mathcal{B}}=0.
\end{equation}
Furthermore, if $c$ is such that $\langle Ac,c\rangle_{\mathcal{B}}>0$ then
we have \[0\leq \left\langle A\left(b-\dfrac{\langle Ab,c\rangle_{\mathcal{B}}}{\langle Ac,c\rangle_{\mathcal{B}}}c\right),b-\dfrac{\langle Ab,c\rangle_{\mathcal{B}}}{\langle Ac,c\rangle_{\mathcal{B}}}c\right\rangle_{\mathcal{B}}=-\dfrac{|\langle Ab,c\rangle_{\mathcal{B}}|^2}{\langle Ac,c\rangle_{\mathcal{B}}},\]
hence \eqref{referee} holds in this case, as well.\\
Thus, $(\ran A,\langle\cdot,\cdot\rangle_A)$ is a
pre-Hilbert
space. We will denote by ${\mathcal H}_A$ its completion and define
$$T\,:\,{\mathcal B}\longrightarrow {\mathcal H}_A,\quad Tb\defi Ab.$$
We have for $b,c\in{\mathcal B}$
\[
\langle T^*(Ac),b\rangle_{{\mathcal B}}= \langle Ac,Tb\rangle_{{\mathcal H}_A} =
\langle Ac,b\rangle_{{\mathcal
B}}.
\]
Hence, $T^*$ extends continuously to the injection map from ${\mathcal H}_A$
into ${\mathcal B}^*$. We note that
\[
\langle Tb,Tc\rangle_{{\mathcal H}_A}=\langle T^*Tb,c\rangle_{{\mathcal B}}
=\langle Ab,c\rangle_{{\mathcal B}}\]
(that is, \eqref{samantha} holds). The claim on the norms is proved as follows: we have
$$\|A\|=\|T^*T\|\le \|T\|^2$$
on the one hand and
\[
\|A\|\ge\sup_{\|b\|=1}\langle Ab,b\rangle_{{\mathcal B}} =\sup_{\|b\|=1} \langle T^*Tb,b\rangle_{{\mathcal B}}
=\sup_{\|b\|=1}\langle Tb,Tb\rangle_{{\mathcal H}_A} =\|T\|^2,
\]
that is, $\|A\|\ge\|T\|^2$ on the other hand.
Combining the two inequalities we obtain \eqref{bagdad}. We now prove
\eqref{sof-sof}. We have:
\[
\begin{split}
|\langle Ab,c\rangle_{\mathcal B}|&=\langle Tb,Tc\rangle_{\mathcal H}\\
&\le \langle Tb,Tb\rangle_{\mathcal H}^{1/2}
\langle Tc,Tc\rangle_{\mathcal H}^{1/2}\\
&=\langle Ab,b\rangle_{\mathcal B}\langle Ac,c\rangle_{\mathcal B}.
\end{split}
\]
\end{proof}
We will say that $A\leq B$ if $B-A\geq 0.$ Note that
\begin{equation}
\label{etoile}
A\leq B \implies \|A\|\leq\|B\|.
\end{equation}
Indeed, from \eqref{bagdad} we have:
$$\|A\|=\sup_{\|b\|=1}\langle Ab,b
\rangle_{{\mathcal B}}\le \sup_{\|b\|=1}\langle Bb,b \rangle_{{\mathcal B}}=\|B\|.$$
\begin{definition}\label{defrk}
Let ${\mathcal H}$ be a Hilbert space of ${\mathcal B}^*$-valued functions
defined on a set $\Omega$ and let $K(z,w)$ be an ${\mathbf L}({\mathcal B},
{\mathcal B}^*)$-valued kernel
defined on $\Omega\times\Omega.$ The kernel $K(z,w)$ is called the reproducing
kernel of the Hilbert space
${\mathcal H}$ if for every $w\in\Omega$ and $b\in{\mathcal B}$
$K(\cdot,w)b\in\mathcal{H}$ and
\[
\langle f, K(\cdot,w)b\rangle_{\mathcal H}= \langle f(w),
b\rangle_{{\mathcal B}},\quad\forall f\in\mathcal{H}.
\]
\end{definition}
\begin{definition}
Let $K(z,w)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
kernel defined on $\Omega\times\Omega.$ The
kernel $K(z,w)$ is said to be positive if for any choice of $z_1,\dots, z_n\in\Omega$ and
$b_1,\dots,b_n\in\mathcal{B}$ it holds that
\[\sum_{j=1}^n\langle K(z_i,z_j)b_j,b_i\rangle_{\mathcal{B}}\geq 0.\]
\end{definition}
\begin{proposition}
\label{p1}
The reproducing kernel $K(z,w)$ of a Hilbert space of ${\mathcal B}^*$-valued functions,
when it exists, is unique and positive.
\end{proposition}
\begin{proposition}
\label{p2}
Let
$K(z,w)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
positive kernel defined on
$\Omega\times\Omega.$ Then there exists a unique Hilbert space of
${\mathcal B}^*$-valued functions defined on
$\Omega$ with the reproducing kernel $K(z,w)$.
\end{proposition}
The proofs of these propositions are the same as in the Hilbert space case
and are therefore omitted.
\begin{remark}\label{brem}
One can derive the notion of a reproducing kernel Hilbert space of
${\mathcal B}$-valued functions from
Definition \ref{defrk} above, using the natural injection
$\tau$ from ${\mathcal B}$ into
${\mathcal B}^{**}$ defined by \eqref{bastille}.
\end{remark} | 2,883 | 13,058 | en |
train | 0.27.2 | \section{Stieltjes integral}
In this section we define the Stieltjes
integral of a scalar function with respect to a ${\mathbf
L}({\mathcal B},{\mathcal B}^*)$--valued positive measure. We
here follows the analysis presented in \cite[\S 4 p. 19]{MR48:904} for
the case of operators in Hilbert spaces. We consider a separable
Banach space ${\mathcal B}$ and an increasing positive function
$$M\,:\, [a,b]\longrightarrow {\mathbf
L}({\mathcal B},{\mathcal B}^*).$$
Thus, $M(t)\geq 0$ for all $t\in[a,b]$ and moreover
$$
a\le t_1\le t_2\le b\Longrightarrow M(t_2)-M(t_1)\ge 0.$$
Let $f(t)$ be a scalar continuous function on $[a,b]$ and let
$$a=t_0\le \xi_1\le t_1\le \xi_2\le t_2\le\cdots\le \xi_{m}\le
t_m=b$$ be a subdivision of $[a,b]$.
The Stieltjes integral $\int_a^b f(t)dM(t)$
is defined to be the limit (in the ${\bf L}({\mathcal B},{\mathcal B}^*)$ topology) of the
sums of the form
$$\sum_{j=1}^m f(\xi_j)(M(t_j)-M(t_{j-1}))$$
as $\sup_{j}|t_j-t_{j-1}|$ goes to $0$.
\begin{theorem}
\label{Sevres-Babylone}
The integral $\int_a^bf(t)dM(t)$ exists.
\end{theorem}
The proof of Theorem \ref{Sevres-Babylone}
is done along the lines of
\cite{MR48:904}. First we need the following lemma.
\begin{lemma}\label{brod}
Let $\alpha_j$ and $\beta_j$ be complex numbers such that
$|\alpha_j|\le |\beta_j|$ ($j=1,2,\ldots m$). Let $H_1,\ldots,
H_m$ be positive operators from ${\mathcal B}$ into ${\mathcal
B}^*$. Then it holds that
\begin{equation*}
\|\sum_{j=1}^m\alpha_j H_j\|\le \|\sum_{j=1}^m |\beta_j|H_j\|.
\end{equation*}
\end{lemma}
\begin{proof} For each $j$ we write $H_j=T_j^*T_j$ where $T_j$ is a
bounded operator from ${\mathcal B}$ into some
Hilbert space ${\mathcal H}_j$ as in Theorem \ref{rosh-hashana}. Then
for $b,c\in{\mathcal B}$ of modulus $1$ we
have:
\[
\begin{split}
|\langle \sum_{j=1}^m\alpha_j H_j b,c\rangle_{{\mathcal B}}| &=
|\sum_{j=1}^m\langle \alpha_j H_j b,c\rangle_{{\mathcal B}}|\\
&\le
\sum_{j=1}^m|\alpha_j| \cdot|\langle T_jb,T_jc\rangle_{{\mathcal H}_j}|\\
&\le \sum_{j=1}^m\sqrt{|\beta_j|} \|T_jb\|_{{\mathcal H}_j}
\sqrt{|\beta_j|}\|T_jc\|_{{\mathcal H}_j}\\
&\le \left(\sum_{j=1}^m|\beta_j|\|T_jb\|^2_{{\mathcal
H}_j}\right)^{1/2}
\left(\sum_{j=1}^m|\beta_j|\|T_jc\|^2_{{\mathcal H}_j}\right)^{1/2}\\
& \le \left(\sum_{j=1}^m|\beta_j|\langle H_jb,
b\rangle_{{\mathcal B}}\right)^{1/2}
\left(\sum_{j=1}^m|\beta_j|\langle H_jc,
c\rangle_{{\mathcal B}}\right)^{1/2}\\
&\le \|\sum_{j=1}^m|\beta_j| H_j\|
\end{split}
\]
where we have used \eqref{bagdad} to get the last inequality.
Thus, taking the supremum on $c$ (of unit norm) we have
$\|\sum_{j=1}^m\alpha_jH_jb\|\le\|\sum_{j=1}^m|\beta_j| H_j\|$,
and hence the required inequality.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Sevres-Babylone}]
It suffices to show that for every $\epsilon>0$ there exists $\delta>0$ such that if
\begin{equation}\label{subdiv}a=t_0\leq t_1\leq\dots\leq t_m=b\end{equation} is a
subdivision of $[a,b]$ such that
$\max_{j}|t_j-t_{j-1}|\leq \delta$ and
$$a=t_0=t_1^0\leq \dots\leq t_1^{k_1}=t_1=t_2^0\leq\dots\leq t_{m}^{k_{m}} =t_m=b$$
is a continuation of the subdivision \eqref{subdiv} then
for every choice of $\xi_j\in[t_{j-1},t_j]$ and
$\xi_j^\ell\in[t_j^{\ell-1},t_j^\ell]$ it holds that
$$\left\|\sum_{j=1}^m f(\xi_j)(M(t_j)-M(t_{j-1}))-\sum_{j=1}^m
\sum_{\ell=1}^{k_j}
f(\xi_j^\ell)(M(t_j^\ell)-M(t_j^{\ell-1}))\right\|\leq\epsilon.$$
Let us take $\delta$ such that
$$|t^\prime-t^{\prime\prime}|\leq\delta\implies
|f(t^\prime)-f(t^{\prime\prime})|\leq\dfrac{\epsilon}{\|M(b)-M(a)\|}.$$
Then the desired conclusion follows from Lemma \ref{brod}.
\end{proof}
\section{Helly's theorem}
The following theorem is proved in the case of {\sl separable} Hilbert space
in \cite{MR48:904} (see Theorem 4.4 p. 22 there). The proof goes in the same
way in the case of separable Banach spaces. We quote it in a version
adapted to the present setting.
\begin{theorem}
\label{helly}
Let $F_n(t)$ ($t\in[0,2\pi]$) be a sequence of positive increasing
${\mathbf L}({\mathcal B},{\mathcal B}^*)$--valued functions such that
$$F_n(t)\le F_0,\quad n=0,1,\ldots\quad\text{and}\quad t\in[0,2\pi],$$
where $F_0$ is some bounded positive
operator. Then, there exists a subsequence of $F_n$
(which we still denote by $F_n$)
which converges weakly for every $t\in[0,2\pi]$. Moreover, for $f(t)$ a
continuous scalar function we have (in the weak sense, and via the
subsequence):
$$
\int_0^{2\pi}f(t)dF(t)=\lim_{n\rightarrow \infty}\int_0^{2\pi}f(t)dF_n(t).
$$
\end{theorem}
The proof given in \cite{MR48:904} relies on the hypothesis of separability
and on the inequalities
\begin{equation}
\label{rastignac}
\begin{split}
|\langle F_n(t)x,y\rangle_{\mathcal B}|&\le\|F_0\|\cdot\|x\|\cdot\|y\|,\quad
x,y\in{\mathcal B}\\
\sum_{\ell=1}^m |
\langle \Delta_{\ell,n}Fx,y\rangle_{\mathcal B}|&\le 2\|F_0\|\cdot
\|x\|\cdot\|y\|
\end{split}
\end{equation}
where $0=t_0\le t_1<\cdots <t_m=2\pi$ and
$\Delta_{\ell,n}F=F_n(t_\ell)-F_n(t_{\ell-1})$.
The first inequality follows from \eqref{etoile}. We prove the second
one using the factorization given in
Theorem \ref{rosh-hashana}. Using this theorem we write:
$$\Delta_{\ell,n}=T_{\ell,n}^*T_{\ell,n}$$
where $T_{\ell,n}$ is a bounded operator from some Hilbert space
${\mathcal H}_{\ell,n}$ into ${\mathcal B}.$
Then, using \eqref{sof-sof} we have:
\[
\begin{split}
\sum_{\ell =1}^m
|\langle\Delta F_{\ell,n}x,y\rangle_{\mathcal B}|&\le
\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}x,x\rangle_{\mathcal B}^{1/2}
\langle\Delta F_{\ell,n}y,y\rangle_{\mathcal B}^{1/2}\\
&\le
\left(\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}x,x\rangle_{\mathcal B}\right)^{1/2}
\left(\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}y,y\rangle_{\mathcal B}\right)^{1/2}\\
&=\langle (F(2\pi)-F(0))x,x\rangle^{1/2}\langle (F(2\pi)-F(0))y,y
\rangle^{1/2}\\
&\le \langle 2F_0x,x\rangle_{\mathcal B}^{1/2}
\langle 2F_0y,y\rangle_{\mathcal B}^{1/2}\\
&\le 2\|F_0\|\cdot \|x\|\cdot\|y\|.
\end{split}
\]
The proof then proceeds as follows (see
\cite[p. 22]{MR48:904}). One applies the first inequality in \eqref{rastignac}
for $x,y$ in a dense countable set $E$ of ${\mathcal B}$. The functions
$t\mapsto \langle F_n(t)x,y\rangle_{\mathcal B}$ are of bounded variation.
An application of the scalar case of Helly's theorem and
the diagonal process allows to find a subsequence of $F_n$ such that
for all $x,y\in E$ and every $t\in[0,2\pi]$ the limit
$$\lim_{n\rightarrow\infty}\langle F_n(t)x,y\rangle_{\mathcal B}$$
exists. We refer the reader to \cite{MR48:904} for more details. | 2,718 | 13,058 | en |
train | 0.27.3 | \section{${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
Carath\'eodory functions}
\begin{definition}
Let $\phi(z)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function,
weakly continuous at the origin
in the sense that
\begin{equation}\label{weak} \langle\phi(z)b,b\rangle_{\mathcal{B}}\rightarrow
\langle\phi(0)b,b\rangle_{\mathcal{B}}\text{ as }z\rightarrow 0,\quad\forall b\in\mathcal{B}.
\end{equation}
For a Carath\'eodory function $\phi(z)$ we shall denote by
$\mathcal{L}(\phi)$ the Hilbert space of $\mathcal{B}^*$-valued
functions with the reproducing kernel $k_\phi(z,w)$.
\end{definition}
We give two representation theorems for Carath\'eodory
functions. In the first we make no assumption on the space ${\mathcal B}$.
Following arguments of Krein
and Langer (see \cite{kl1}), we prove the existence of a realization of the
form \eqref{anaelle2}. The second theorem assumes that
the space ${\mathcal B}$ is separable. We prove that in this case
the Carath\'eodory functions can be characterized as functions
analytic in the open unit disk with positive real part. Then we
derive a Herglotz-type representation formula.
\begin{theorem} Let $\Omega$ be a neighborhood of the origin and
let $\phi(z)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function
defined in $\Omega$ and weakly continuous at
the origin (in the sense \eqref{weak}).
Then $\phi(z)$ is a Carath\'eodory function if and only if it admits
the representation
\begin{equation*}
\phi(z)^*\big|_{\mathcal B}=D+C^*(I-z^*V)^{-1}(I+z^*V)C,\quad z\in\Omega,
\end{equation*}
or equivalently,
\begin{equation}
\label{sarah}
\phi(z)=D^*\big|_{\mathcal B}+C^*(I+zV^*)(I-zV^*)^{-1}C,\quad z\in\Omega,
\end{equation}
where $V$ is an isometric operator in some Hilbert space
${\mathcal H}$, $C$ is a bounded operator from ${\mathcal
B}$ into ${\mathcal H}$ and $D$ is a purely imaginary
operator from ${\mathcal B}$ into ${\mathcal B}^{*}$ in the sense that
\begin{equation}
\label{D1}
D+D^*\big|_{{\mathcal B}}=0.
\end{equation}
In particular, every Carath\'eodory function has an analytic extension to
the whole open unit disk.
\label{Ivry-sur-Seine}
\end{theorem}
\begin{proof}
Let $\phi(z)$ be a Carath\'eodory function. First we observe that elements of
$\mathcal{L}(\phi)$ are weakly continuous at the origin:
\[\langle f(w),b\rangle_{\mathcal{B}}\rightarrow
\langle f(0),b\rangle_{\mathcal{B}}\text{ as }w\rightarrow 0,\quad
\forall\,f\in\mathcal{L}(\phi),b\in\mathcal{B}.\]
Indeed, this is a consequence of the Cauchy -- Schwarz inequality as
\[\langle f(w),b\rangle_{\mathcal{B}}-
\langle f(0),b\rangle_{\mathcal{B}}=
\langle f,(k_\phi(\cdot,w)-k_\phi(\cdot,0))b\rangle_{\mathcal{L}(\phi)}\]
and
\[
\|(k_\phi(\cdot, w)-k_\phi(\cdot,0))b\|^2_{\mathcal{L}(\phi)}
=\frac{|w|^2}{1-|w|^2}\Re \langle \phi(w)b,b\rangle_{{\mathcal B}}.
\]
We consider in ${\mathcal L}(\phi)\times{\mathcal L}(\phi)$ the linear
relation $R$ spanned by the pairs
\[
R=
\left(\begin{array}{cc}
\sum k_\phi(z,w_i)w_i^*b_i,\sum k_\phi(z,w_i)b_i-k_\phi(z,0)(\sum b_i)
\end{array}\right)
\]
where all the $b_i\in{\mathcal B}$, the $w_i\in\Omega$ and where all the
sums are finite. This
relation is densely defined because of the weak continuity of the elements of
${\mathcal L}(\phi)$ at the origin.
Indeed, let $f\in{\mathcal L}(\phi)$ be orthogonal to the domain of $R$. Then,
\[
\langle f, k_\phi(\cdot,w)b\rangle_{{\mathcal L}(\phi)}=\langle f(w),b
\rangle_{{\mathcal B}}=0\]
for all $b\in{\mathcal B}$ and all
points $w\not =0$ in the domain of $f$. Thus $f(w)=0$ at all these points $w$
and the continuity hypothesis implies that also $f(0)=0$.
The relation $R$ is readily seen to be isometric. Its closure is thus the
graph of an isometry, which we call $V$. We have:
\[
V(k_\phi(z,w)w^*b)=k_\phi(z,w)b-k_\phi(z,0)b,\]
and in particular
\begin{equation}
\label{Alfort-Ecole-veterinaire}
(I-w^*V)^{-1}k_\phi(\cdot, 0)b=k_\phi(\cdot, w)b.
\end{equation}
Denote by $C$ the map
$$C\,:\,{\mathcal B}\longrightarrow {\mathcal L}(\phi),\quad(Cb)(z)
\defi k_\phi(z,0)b.$$
Then, for $f\in{\mathcal L}(\phi)$, $C^*f=f(0)$.
Applying $C$ on the left on both sides of
\eqref{Alfort-Ecole-veterinaire} we obtain
\[\frac{\phi(0)+\phi(w)^*\big|_{\mathcal B}}{2}b=C^*(I-w^*V)^{-1}Cb.\]
Since $$C^*Cb=\frac{\phi(0)+\phi(0)^*\big|_{\mathcal B}}{2}b$$
we obtain
\[
\begin{split}
\phi(0)+\phi(w)^*\big|_{\mathcal B}&=2C^*(I-w^*V)^{-1}C-C^*C+C^*C\\
&=C^*(I-w^*V)^{-1}(I+w^*V)C+C^*C
\end{split}
\]
so that
\[
\phi(w)^*\big|_{{\mathcal B}}+\frac{\phi(0)-\phi(0)^*\big|_{\mathcal B}}{2}
=
C^*(I-w^*V)^{-1}(I+w^*V)C,
\]
which gives the required formula with
\begin{equation}
\label{D} D=\frac{\phi(0)-\phi(0)^*\big|_{\mathcal B}}{2}.
\end{equation}
We now prove the converse statement and first compute
$$\langle k_\phi(z,w)x,y\rangle_{{\mathcal B}}
\quad{\rm for}\quad x,y\in{\mathcal B}.$$
We have
\[
\begin{split}
\langle\phi(w)^*\big|_{\mathcal B}x,y\rangle_{{\mathcal B}}&=
\langle Dx,y\rangle_{{\mathcal B}}+\langle
(I-w^*V)^{-1}(I+w^*V)Cx,Cy\rangle_{{\mathcal L}(\phi)}.
\end{split}
\]
We have, with $\tau$ the natural injection from ${\mathcal B}$ into
${\mathcal B}^{**}$ (see \eqref{bastille}):
\begin{equation}
\begin{split}
\langle \phi(z)x,y\rangle_{{\mathcal B}} &= \overline{\langle \tau
y ,\phi(z)x\rangle_{{\mathcal B}^*}}\\
&=\overline{\langle \phi(z)^*\tau y,x\rangle_{{\mathcal B}^*}}\\
&= \overline{\langle \phi(z)^*\big|_{\mathcal B}y,x
\rangle_{{\mathcal B}}}\\
&=\overline{\langle D y,x\rangle_{{\mathcal B}}}+
\langle(I-zV^*)^{-1}(I+zV^*)Cx,Cy\rangle_{{\mathcal L}(\phi)}\\
&=\langle D^*\big|_{\mathcal B}x ,y\rangle_{{\mathcal B}}+
\langle(I-zV^*)^{-1}(I+zV^*)Cx,Cy\rangle_{{\mathcal L}(\phi)},
\end{split}
\label{elinor}
\end{equation}
and so
$$\langle k_\phi(z,w)x,y\rangle_{{\mathcal B}}
=\langle (I-zV^*)^{-1}Cx,(I-wV^*)^{-1}Cy\rangle_{{\mathcal L}(\phi)},$$
from which follows the positivity of $k_\phi(z,w)$. Finally,
\eqref{elinor} also implies \eqref{sarah} and this concludes the proof.
\end{proof} | 2,289 | 13,058 | en |
train | 0.27.4 | \begin{remark}
Although the above argument is very close to the one in
\cite[p. 365--366]{kl1} we note the following: we use a concrete
space (the space ${\mathcal L}(\phi)$) to build the relation rather than
abstract elements and the relation $R$ is defined slightly differently.
\end{remark}
\begin{remark}\label{indef}
As already mentioned, the above argument still goes through when the
kernel has a finite number of negative squares. In this case the space
$\mathcal{L}(\phi)$ is a Pontryagin space. For $b\in
{\mathcal B}$ and sufficiently small $h\in{\mathbb C}$ we consider
the functions $f_h(z)=(K(z,w+h)-K(z,w))b$, which have the following
properties:
\begin{align}
\nonumber
\lim_{h\rightarrow 0}\langle f_h,f_h\rangle_{{\mathcal L}(\phi)}&=0,\\
\lim_{h\rightarrow 0}\langle f,f_h\rangle_{{\mathcal L}(\phi)}&=0,
\quad\forall f\in\spa\{k_\phi(\cdot,w)b\}.
\nonumber
\end{align}
It follows from the convergence criteria in Pontryagin spaces (see
\cite{ikl}, \cite[p. 356]{kl1}) that
$$\lim_{h\rightarrow 0}\langle f,f_h\rangle_{{\mathcal L}(\phi)}=0,\quad
\forall f\in\mathcal{L}(\phi).$$ The fact that the relation is the graph of
an isometric operator is
proved in \cite[Theorem 1.4.2 p. 29]{adrs}. This follows from a theorem of
Shmulyan which states that a
contractive relation between Pontryagin spaces of same index is the graph
of a contractive operator; see
\cite{s} and \cite[Theorem 1.4.1 p. 27]{adrs}.
\end{remark}
\begin{theorem}
Let ${\mathcal B}$ be a separable Banach space and let $\phi(z)$ be a
${\mathbf L}({\mathcal B},{\mathcal
B}^*)$--valued function analytic in the open unit disk, such that
\[\phi(z)+{\phi(z)^*}\bigm|_{ \mathcal{B}}\,
\geq 0.\] Then there exists an increasing ${\mathbf L}({\mathcal B},{\mathcal B}^*)$--valued
function $M(t)$
($t\in[0,2\pi]$) and a purely imaginary operator $D$ (that is, satisfying \eqref{D1})
such that
\[
\phi(z)=D+\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}dM(t),
\]
where the integral is defined in the weak sense. Furthermore the
kernel $k_\phi(z,w)$ is positive in ${\mathbb D}$.
\label{Antony}
\end{theorem}
\begin{proof} We follow the arguments in \cite{MR48:904}, and will apply
Theorem \ref{helly}. The separability hypothesis of ${\mathcal B}$ is used at
this point.\\
We first assume that $\phi(z)$
is analytic in $|z|<1+\epsilon$ with $\epsilon>0$. We have
(the existence of the integrals follows from Theorem
\ref{Sevres-Babylone}):
\[
\begin{split}
\frac{1}{4\pi}
\int_0^{2\pi}\phi(e^{it})\frac{e^{it}+z}{e^{it}-z}dt&=\phi(z)-\frac{\phi(0)}{2}\\
\frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{it})^*\big|_{\mathcal B}
\frac{e^{it}+z}{e^{it}-z}dt&= \frac{1}{2\pi
}\int_0^{2\pi}(\phi(e^{-it})^*\big|_{\mathcal B}
\frac{e^{-it}+z}{e^{-it}-z}\\
&=\frac{(\phi(0))^*\big|_{\mathcal B}}{2}.
\end{split}
\]
Thus,
\[
\phi(z)=D+\int_0^{2\pi}\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{e^{it}+z}{e^{it}-z}dt,
\] with $D$ as in \eqref{D} and the formula for general
$\phi(z)$ follows from Helly's theorem applied to the measures
$$
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}dt,\quad r<1$$
(or more precisely to a sequence $r_n\rightarrow 1$).\\
We now prove the positivity of the kernel $k_\phi(z,w)$ and first
assume that $\phi(z)$ is analytic in $|z|<1+\epsilon$ as above.
We have:
\[
\frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{-it}))^*\big|_{\mathcal B}
\frac{e^{it}+z}{e^{it}-z}dt=(\phi(z^*))^*\big|_{\mathcal B}
-\frac{(\phi(0))^*\big|_{\mathcal B}}{2}.
\]
Thus
\[
(\phi(z^*))^*\big|_{\mathcal B} -\frac{(\phi(0))^*\big|_{\mathcal
B}}{2}= \frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{it}))^*\big|_{\mathcal
B} \frac{e^{-it}+z}{e^{-it}-z}dt
\]
and so
$$
(\phi(z^*))^*\big|_{\mathcal B}=
-D+\int_0^{2\pi}\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{1+ze^{it}}{1-ze^{it}}dt
$$
since
$$
\frac{1}{4\pi}\int_0^{2\pi}\phi(e^{it})\frac{1+ze^{it}}{1-ze^{it}}dt
=
\frac{1}{4\pi}\int_0^{2\pi}\phi(e^{-it})\frac{e^{it}+z}{e^{it}-z}dt=\frac{\phi(0)}{2}.$$
Thus,
$$k_\phi(z,w)=\int_0^{2\pi}
\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal B}\right)}{4\pi}\frac{1}{(e^{it}-z)(e^{it}-w)^*}dt.$$
The positivity follows. The case of general $\phi(z)$ is done by approximation using Helly's theorem; indeed
using Theorem \ref{helly} we have for general $\phi(z)$:
\[
\begin{split}
\langle k_\phi(w_\ell,w_j)b_j,b_\ell\rangle_{\mathcal B}&=
\\
&\hspace{-2cm}=
\lim_{r\rightarrow 1}\int_0^{2\pi}
\left\langle
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{1}{(e^{it}-w_\ell)(e^{it}-w_j)^*}b_j,b_\ell
\right\rangle_{\mathcal B}dt\\
&\hspace{-2cm}=
\lim_{r\rightarrow 1}\int_0^{2\pi}
\left\langle
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{b_j}{(e^{it}-w_j)^*},\frac{b_\ell}{(e^{it}-w_\ell)^*}
\right\rangle_{\mathcal B}dt.
\end{split}
\]
\end{proof} | 1,977 | 13,058 | en |
train | 0.27.5 | \section{The case of ${\mathbf L}({\mathcal B}^{*},{\mathcal B})$--valued
functions} We turn to the case of ${\mathbf L}({\mathcal
B}^{*},{\mathcal B})$--valued functions.
Using the natural
injection $\tau$
$$\mathcal{B}\stackrel{\tau}{\mapsto}\mathcal{B}^{**}$$
defined by \eqref{bastille} we shall say that an ${\mathbf
L}({\mathcal B}^{*},{\mathcal B})$--valued function $\phi(z)$ is
a Carath\'eodory function if the ${\mathbf L}({\mathcal
B}^{*},{\mathcal B}^{**})$-valued function $\tau\phi(z)$ is a
Carath\'eodory function.
\begin{theorem} An ${\mathbf L}({\mathcal B}^*,{\mathcal B})$--valued
function $\phi(z)$ defined in a neighborhood of the origin
and weakly continuous at the origin is a Carath\'eodory function
if and only if it admits the representation
\begin{equation*}
\phi(z)^*=D+C^*(I-z^*V)^{-1}(I+z^*V)C,
\end{equation*}
or, equivalently,
\[
\tau\phi(z)=D^*\big|_{{\mathcal B}_*}+C^*(I+zV^*)(I-zV^*)^{-1}C,\]
where $V$ is an isometric operator in some Hilbert space
${\mathcal H}$, where $C$ is a bounded operator from ${\mathcal
B}^*$ into ${\mathcal H}$ and where $D$ is a purely imaginary
operator from ${\mathcal B}^*$ into ${\mathcal B}^{**}$.
In particular $\phi(z)$ has an analytic extension to the whole open unit disk.
\end{theorem}
\begin{proof} By Theorem \ref{Ivry-sur-Seine} (with $\mathcal{B}$ replaced by
$\mathcal{B}^*$),
$\phi(z)$ is a Carath\'eodory function if and only if
\[(\tau\phi(z))^*\bigm|_{\mathcal{B}^*}=D+C^*(I-z^*V)^{-1}(I+z^*V)C,\]
where $C, V, D$ have the stated properties.
But $(\tau\phi(z))^*\bigm|_{\mathcal{B}^*}=\phi(z)^*.$
\end{proof}
\end{document} | 573 | 13,058 | en |
train | 0.28.0 | \begin{document}
\title{Most incompatible measurements for robust steering tests}
\author{Jessica Bavaresco}
\email{[email protected]}
\affiliation{Departamento de F\'isica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\affiliation{Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria}
\author{Marco T\'ulio Quintino}
\affiliation{Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan}
\author{Leonardo Guerini}
\affiliation{Departamento de Matem\'atica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\affiliation{ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain}
\author{Thiago O. Maciel}
\affiliation{Departamento de F\'isica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\author{Daniel Cavalcanti}
\affiliation{ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain}
\author{Marcelo Terra Cunha}
\affiliation{Departamento de Matem\'atica Aplicada, IMECC-Unicamp, 13084-970, Campinas, S\~ao Paulo, Brazil}
\date{\today}
\begin{abstract}
We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to $N$ measurements of $k$ outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a $d$-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements.
We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering isotropic states and provide candidates to the most incompatible sets of measurements in each case.
Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
Correlations arising from local measurements on entangled states can lead to statistics that cannot be explained by any local causal theory \cite{RevNL}. This nonlocal aspect of quantum mechanics can be analyzed from two nonequivalent perspectives, Bell nonlocality \cite{Bell} and Einstein-Podolsky-Rosen (EPR) steering \cite{EPR,Wiseman}. On a bipartite scenario, while Bell nonlocality deals with a full device-independent approach where a correlation experiment is analyzed only considering the probability relations between inputs and outcomes, EPR steering plays an intermediate role between entanglement and Bell nonlocality by doing a device-independent analysis only on one side of the experiment while treating the other side in a device-dependent manner (e.g., performing full state tomography).
Although the first notions of EPR steering date back to 1935 \cite{EPR}, its modern mathematical formulation only appeared in 2007 \cite{Wiseman}, and many of its fundamental properties \cite{MTGeneralMeas,MTOneWay,MultipartiteDani,aolita14,quintino17} and applications to semi-device independent protocols \cite{tomamichel11,tomamichel12,OneSidedDIQC,law14,passaro15,kogias17} are only being understood recently. In order to get a better understanding of EPR steering and make use of its practical applications, an important task is to determine which states can lead to these nonlocal correlations.
EPR steering can be certified with the use of steering witnesses \cite{SteIneq}, but finding suitable inequalities and choosing appropriate measurements to reveal this sort of nonlocality of a given quantum state remains a nontrivial task. On the other hand, proving that a quantum state cannot lead to EPR steering can be done by constructing a local hidden state (LHS) model that is able to simulate the statistics of the quantum state \cite{WernerModel,Wiseman,sania14,JoeCriterion}. Recently, a general algorithm to construct LHS models for quantum states was presented \cite{LHScavalcanti,LHShirsch}, regarding scenarios where all (i.e., infinitely many) measurements are considered. However, from a practical perspective, it is important to characterize what one can do with a limited number of measurements and outcomes, or yet, when even the structure of the allowed measurements is restricted.
Since the ability to demonstrate steering is intimately related to the ability to perform incompatible measurements \cite{krausbook,buschbook}, by addressing the problem of characterizing the steerability of quantum states under restrictive scenarios one can simultaneously address the problem of characterizing the ability to jointly perform a set of unknown measurements subjected to the same restrictions \cite{MTJM,UolaJM1}. Although the question of whether a set of fixed (known) measurements is jointly measurable can be decided by semidefinite programming (SDP) \cite{wolf09} -- which is also the case for deciding whether a given quantum state is steerable when subjected to a set of fixed (known) measurements \cite{MattPusey,SteeringWeight} -- if the complete description of the measurements is not known, there do not exist general methods to characterize steerability or joint measurability.
In this paper we consider steering and joint measurability in scenarios where the number of measurements and outcomes is finite. By systematically applying adaptations of the parametric search, the seesaw algorithm \cite{RevSDP}, and the outer polytope approximation \cite{SuperPOVM}, we derive upper and lower bounds to the maximal amount of white noise that a quantum state can endure before it is no longer able to demonstrate steering when subjected to a set of $N$ general local measurements with $k$ outcomes. Using the same methods, we calculate upper and lower bounds for the minimum amount of white noise that must be applied to \textit{any} set of general $N$ qudit $k$-outcome measurements so that it is assured that they can be jointly performed. By imposing further restrictions on the measurement scenarios, we also study prominent classes of measurements that are known to be useful in many quantum information tasks, such as projective measurements, symmetric informationally complete (SIC) measurements \cite{renes04}, and measurements from mutually unbiased bases (MUBs) \cite{durt10}.
We present our calculations for qubit states in scenarios ranging from $2$ to $18$ projective, planar projective, symmetric, and general measurements and provide strong evidence that, in the considered scenarios, general positive-operator valued measures (POVMs) do not outperform projective measurements for exhibiting steering. We also show that our optimal sets of qubit measurements are not distributed in the Bloch sphere according to any of the most intuitive patterns, such as the vertices of Platonic solids, the distribution of electrons on a sphere in the Thomson problem \cite{Thomson}, and the Fibonacci spiral \cite{Fibonacci}. We present an alternative candidate for this distribution for the cases of $N\in\{2,\ldots,6\}$ measurements, supported by our numerical findings. For higher dimensions, we present evidence that increasing the number of outcomes beyond the value of the local dimension of the state does not improve white-noise robustness, again implying that projective measurements are optimal for steering. We also prove that, in many cases, incomplete sets of MUB measurements are not optimal, while providing numerical evidence that complete sets may be optimal for steering isotropic states. | 2,033 | 18,845 | en |
train | 0.28.1 | \section{Introduction}\label{intro}
Correlations arising from local measurements on entangled states can lead to statistics that cannot be explained by any local causal theory \cite{RevNL}. This nonlocal aspect of quantum mechanics can be analyzed from two nonequivalent perspectives, Bell nonlocality \cite{Bell} and Einstein-Podolsky-Rosen (EPR) steering \cite{EPR,Wiseman}. On a bipartite scenario, while Bell nonlocality deals with a full device-independent approach where a correlation experiment is analyzed only considering the probability relations between inputs and outcomes, EPR steering plays an intermediate role between entanglement and Bell nonlocality by doing a device-independent analysis only on one side of the experiment while treating the other side in a device-dependent manner (e.g., performing full state tomography).
Although the first notions of EPR steering date back to 1935 \cite{EPR}, its modern mathematical formulation only appeared in 2007 \cite{Wiseman}, and many of its fundamental properties \cite{MTGeneralMeas,MTOneWay,MultipartiteDani,aolita14,quintino17} and applications to semi-device independent protocols \cite{tomamichel11,tomamichel12,OneSidedDIQC,law14,passaro15,kogias17} are only being understood recently. In order to get a better understanding of EPR steering and make use of its practical applications, an important task is to determine which states can lead to these nonlocal correlations.
EPR steering can be certified with the use of steering witnesses \cite{SteIneq}, but finding suitable inequalities and choosing appropriate measurements to reveal this sort of nonlocality of a given quantum state remains a nontrivial task. On the other hand, proving that a quantum state cannot lead to EPR steering can be done by constructing a local hidden state (LHS) model that is able to simulate the statistics of the quantum state \cite{WernerModel,Wiseman,sania14,JoeCriterion}. Recently, a general algorithm to construct LHS models for quantum states was presented \cite{LHScavalcanti,LHShirsch}, regarding scenarios where all (i.e., infinitely many) measurements are considered. However, from a practical perspective, it is important to characterize what one can do with a limited number of measurements and outcomes, or yet, when even the structure of the allowed measurements is restricted.
Since the ability to demonstrate steering is intimately related to the ability to perform incompatible measurements \cite{krausbook,buschbook}, by addressing the problem of characterizing the steerability of quantum states under restrictive scenarios one can simultaneously address the problem of characterizing the ability to jointly perform a set of unknown measurements subjected to the same restrictions \cite{MTJM,UolaJM1}. Although the question of whether a set of fixed (known) measurements is jointly measurable can be decided by semidefinite programming (SDP) \cite{wolf09} -- which is also the case for deciding whether a given quantum state is steerable when subjected to a set of fixed (known) measurements \cite{MattPusey,SteeringWeight} -- if the complete description of the measurements is not known, there do not exist general methods to characterize steerability or joint measurability.
In this paper we consider steering and joint measurability in scenarios where the number of measurements and outcomes is finite. By systematically applying adaptations of the parametric search, the seesaw algorithm \cite{RevSDP}, and the outer polytope approximation \cite{SuperPOVM}, we derive upper and lower bounds to the maximal amount of white noise that a quantum state can endure before it is no longer able to demonstrate steering when subjected to a set of $N$ general local measurements with $k$ outcomes. Using the same methods, we calculate upper and lower bounds for the minimum amount of white noise that must be applied to \textit{any} set of general $N$ qudit $k$-outcome measurements so that it is assured that they can be jointly performed. By imposing further restrictions on the measurement scenarios, we also study prominent classes of measurements that are known to be useful in many quantum information tasks, such as projective measurements, symmetric informationally complete (SIC) measurements \cite{renes04}, and measurements from mutually unbiased bases (MUBs) \cite{durt10}.
We present our calculations for qubit states in scenarios ranging from $2$ to $18$ projective, planar projective, symmetric, and general measurements and provide strong evidence that, in the considered scenarios, general positive-operator valued measures (POVMs) do not outperform projective measurements for exhibiting steering. We also show that our optimal sets of qubit measurements are not distributed in the Bloch sphere according to any of the most intuitive patterns, such as the vertices of Platonic solids, the distribution of electrons on a sphere in the Thomson problem \cite{Thomson}, and the Fibonacci spiral \cite{Fibonacci}. We present an alternative candidate for this distribution for the cases of $N\in\{2,\ldots,6\}$ measurements, supported by our numerical findings. For higher dimensions, we present evidence that increasing the number of outcomes beyond the value of the local dimension of the state does not improve white-noise robustness, again implying that projective measurements are optimal for steering. We also prove that, in many cases, incomplete sets of MUB measurements are not optimal, while providing numerical evidence that complete sets may be optimal for steering isotropic states.
\section{Preliminaries}
\subsection{Einstein-Podolsky-Rosen steering}
Bipartite steerability is usually defined in terms of an assemblage. Let $\rho_{AB}$ be a bipartite quantum state shared by Alice and Bob and let $\{M_{a|x}\}$ be a set of measurements on Alice's subsystems. Then, an assemblage $\{\sigma_{a|x}\}$ is defined as
\begin{equation}
\sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}\,\rho_{AB}),
\end{equation}
for all $a,x$, where $x\in\{1,\ldots,N\}$ and $a\in\{1,\ldots,k\}$ label Alice's measurements and outcomes, respectively, and $\text{\normalfont Tr}_A$ denotes the partial trace over the Hilbert space of Alice. An assemblage does not demonstrate steering when it admits an LHS model, namely, when there exists $\Lambda$ such that
\begin{equation}\label{eqlhs}
\sigma_{a|x} = \sum_\lambda \pi(\lambda)p_A(a|x,\lambda)\rho_\lambda,
\end{equation}
for all $a,x$, where $\lambda\in\Lambda$ are the possible values that can be assumed by a local hidden variable with probability $\pi(\lambda)$, $p_A(a|x,\lambda)$ is the probability of Alice's obtaining outcome $a$ conditioned on her choice of measurement $x$ and $\lambda$, and, finally, $\rho_\lambda$ is a local hidden state held by Bob that is conditioned by the value $\lambda$ and independent of Alice's measurements and outcomes. An assemblage demonstrates steering when it does not admit such decomposition \cite{Wiseman} or, equivalently, when it violates a steering inequality \cite{SteIneq}. A quantum state $\rho_{AB}$ is unsteerable if all assemblages that can be generated by performing local measurements on it admit an LHS model. On the other hand, a quantum state is steerable if there exists a set of measurements that, when locally performed on it, generates an assemblage that violates a steering inequality.
\subsection{Measurement incompatibility}
A set of measurements $\{M_{a|x}\}$, where $x\in\{1,\ldots,N\}$ labels the measurements in the set and $a\in\{1,\ldots,k\}$ labels the outcomes of each measurement, is jointly measurable, or compatible, if there exists a \textit{joint measurement}, $\{M_\lambda\}$, such that
\begin{equation}
M_{a|x} = \sum_\lambda \pi(\lambda) p(a|x,\lambda) M_\lambda,
\end{equation}
for all $a,x$, where $\pi(\lambda)$ and $p(a|x,\lambda)$ are elements of probability distributions. Hence, all POVM elements $M_{a|x}$ can be recovered by coarse-graining over the joint measurement $\{M_\lambda\}$.
Although for projective measurements joint measurability is equivalent to commutation, general POVMs from a jointly measurable set may not commute \cite{kru,teiko08}. In this sense, joint measurability is a more general definition of incompatibility.
\subsection{Main problem} \label{Quantify}
Consider the depolarizing map $\Lambda_\eta$ acting on the Hermitian operator $A$ of a $d$-dimensional Hilbert space $\set{H}$, defined as
\begin{equation}\label{eqdepolmap}
A\mapsto\Lambda_{\eta}(A) = \eta A+(1-\eta)\text{\normalfont Tr}(A)\frac{\mathbb{1}}{d}.
\end{equation}
The depolarizing map can be physically interpreted as the effect of the presence of white noise in the implementation of $A$.
When applied to elements of an assemblage it defines a steering quantifier, the white-noise robustness of an assemblage,
\begin{equation}
\eta(\{\sigma_{a|x}\})=\max\left\{\eta \ | \ \{\Lambda_{\eta}(\sigma_{a|x})\}_{a,x} \in LHS\right\},
\end{equation}
where $LHS$ is the set of assemblages that admit an LHS model and, hence, do not demonstrate steering. Therefore, $\eta(\{\sigma_{a|x}\})$ is the exact value of $\eta$, called the critical visibility of the assemblage, above which $\{\sigma_{a|x}\}$ no longer admits an LHS model. Since $\{\Lambda_1(\sigma_{a|x})\}$ is the assemblage itself and $\{\Lambda_0(\sigma_{a|x})\}$ is such that each of its elements corresponds to a multiple of the identity (and therefore it always admits an LHS model), by convexity it is guaranteed that the critical visibility of the assemblage $\eta(\{\sigma_{a|x}\})$ lies in $[0,1]$.
Given an assemblage, its critical visibility can be calculated by an SDP (see Sec. \ref{upper} and see Ref. \cite{RevSDP} for a review of SDP characterization of steering). Similarly, by applying the depolarizing map to a set of measurements $\{M_{a|x}\}$ instead of an assemblage, one can define the critical visibility for a set of measurements to be incompatible, i.e., a value of $\eta$ above which a set of measurements can no longer be described by a joint POVM.
Here we are interested in calculating the minimum of the quantity $\eta(\{\sigma_{a|x}\})$ among all the possible choices of $N$ measurements with $k$ outputs for a fixed quantum state $\rho_{AB}$. Formally this quantity can be defined as
\begin{equation}
\begin{split}
\eta^*(\rho_{AB},N,k) = \min_{\{M_{a|x}\}} \Big\{ &\eta(\{\sigma_{a|x}\}) \ | \\
&\sigma_{a|x}=\text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}\,\rho_{AB}) \Big\},
\end{split}
\end{equation}
where the minimization runs over sets $\{M_{a|x}\}$ of $N$ $k$-outcome measurements. The value $\eta^*(\rho_{AB}, N, k)$ is the critical visibility of the quantum state $\rho_{AB}$ when subjected to $N$ measurements of $k$ outcomes. Note that for $\eta\leq\eta^*(\rho_{AB}, N, k)$, the state $\rho_{AB}$ is unsteerable for {\it all} sets of $N$ $k$-outcome measurements, and for $\eta>\eta^*(\rho_{AB}, N, k)$, $\rho_{AB}$ is steerable for {\it at least one} set of $N$ $k$-outcome measurements.
Unlike the critical visibility of an assemblage $\eta(\{\sigma_{a|x}\})$, the critical visibility of a quantum state $\eta^*(\rho_{AB}, N, k)$ is the solution of a min-max optimization problem and cannot be calculated by an SDP. In this work we provide methods to obtain upper and lower bounds for $\eta^*(\rho_{AB},N,k)$.
\subsection{Connection to the most incompatible measurements}\label{JM=STE}
In Refs. \cite{MTJM,UolaJM1} the authors have proved that a set of measurements $\{M_{a|x}\}$ is not jointly measurable \textit{if and only if} Alice can steer Bob by performing the same measurements on her share of a maximally entangled state $\ket{\phi_d^+}:=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}$, where $d$ stands for the local dimension of the quantum system. Hence, the critical visibility $\eta^*(\ket{\phi_d^+}, N,k)$ coincides with the critical visibility for which \textit{any} set of $N$ measurements with $k$ outcomes is jointly measurable.
Following from the definition of the depolarizing map (Eq. (\ref{eqdepolmap})) and the maximally entangled states, it is easy to show that the noisy assemblage $\{\Lambda_\eta(\sigma_{a|x})\}$, resulting from applying the depolarizing map to an assemblage generated by performing local measurements $\{M_{a|x}\}$ on a maximally entangled state $\ket{\phi_d^+}$, is equivalent to the assemblage resulting from locally performing measurements $\{M_{a|x}\}$ on the noisy state $(\mathbb{1}\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})$. Namely,
\begin{equation}
\Lambda_\eta(\sigma_{a|x})=\text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}_d\,(\mathbb{1}_d\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})),
\end{equation}
where
\begin{equation}
(\mathbb{1}_d\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})=\eta\ketbra{\phi_d^+}{\phi_d^+}+(1-\eta)\frac{\mathbb{1}_{d^2}}{d^2}
\end{equation}
is the isotropic state of local dimension $d$. Therefore, the critical visibility of the maximally entangled states $\eta^*(\ket{\phi_d^+},N,k)$ is equal to the critical value of the parameter $\eta$ of the isotropic states for which they can demonstrate steering, which, in turn, is equal to the critical visibility for any set of $N$ unknown qudit measurements with $k$ outcomes to be compatible. For this reason we speak equivalently of the critical visibility of the maximally entangled states, isotropic states, and joint measurability. To simplify notation, we define this quantity as $\eta^*(d, N,k)\coloneqq\eta^*(\ket{\phi_d^+}, N,k)$.
For general states, one can lower-bound the noise robustness of joint measurability by that of steerability \cite{QuantifiersDani}. | 3,690 | 18,845 | en |
train | 0.28.2 | \section{Methods}\label{methods}
In the following we describe three methods we used to characterize the steerability of quantum states subjected to restricted measurement scenarios. The first method provides upper bounds for $\eta^*(\rho_{AB}, N,k)$ in scenarios where not only the number of measurements and outcomes is fixed but possibly also the structure of the POVMs. The second one provides upper bounds for $\eta^*(\rho_{AB},N,k)$ when only the number of measurements and outcomes is fixed (considering general measurements). Both methods provide candidates for the optimal set of measurements in a given scenario. The third method provides lower bounds for $\eta^*(\rho_{AB},N,k)$ and constructs LHS models for quantum states when the number of measurements and outcomes is fixed.
All code used in this work is available in Ref. \cite{Code}.
\subsection{Upper bounds for $\eta^*(\rho_{AB},N,k)$}\label{upper}
\textit{Search algorithm}. For a given quantum state $\rho_{AB}$ and a fixed set of measurements $\{M_{a|x}\}$, the critical visibility $\eta(\{\sigma_{a|x}\})$ of the assemblage $\{\sigma_{a|x}\}$, which is generated by locally performing these measurements on the given state, can be calculated by SDP
\begin{align}
\begin{split}\label{wnrsdp}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{M_{a|x}\} \\
\max &\hspace{0.2cm} \eta \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}),\ \forall\, a,x \\
& \hspace{0.2cm} \eta\sigma_{a|x}+(1-\eta)\text{\normalfont Tr}(\sigma_{a|x})\frac{\mathbb{1}}{d}=\sum_\lambda D(a|x,\lambda)\sigma_\lambda,\ \forall\, a,x \\
& \hspace{0.2cm} \sigma_\lambda \geq 0, \ \forall\,\lambda,
\end{split}
\end{align}
where $D(a|x,\lambda)$ are elements of deterministic probability distributions and $\lambda\in\{1,\ldots,k^N\}$.
For a fixed quantum state $\rho_{AB}$, different sets of measurements can be tested, each set requiring one SDP to calculate the value of $\eta(\{\sigma_{a|x}\})$. The first method we propose is to parametrize the sets of measurements allowed in a given scenario and, by varying these parameters, explore the solution of multiple SDPs to calculate a bound for $\eta^*(\rho_{AB},N,k)$.
Two important facts can be explored to facilitate this task. The first one is that it is only necessary to optimize over extremal measurements. This is due to the fact that the critical value of $\eta$ depends linearly on the choice of measurements, hence, by convexity, the optimal value will be obtained over extremal measurements. The second fact is that for a system of dimension $d$, extremal measurements have at most $d^2$ outcomes \cite{ExtremalPOVM}.
Aside from the restriction on the number of measurements and outcomes, it is possible to impose restrictions on the parametrization that specify a certain kind of measurement that can be more relevant to the problem one wishes to approach. For instance, it is possible to perform an optimization over only projective measurements or other POVMs with some specific structure (e.g., SIC-POVMs).
The optimization tools chosen for this work are the MATLAB functions \code{fminsearch} \cite{fminsearch}, an unconstrained nonlinear multivariable optimization tool, and \code{fmincon} \cite{fmincon}, a constrained nonlinear multivariable optimization tool. These methods are heuristic and, as such, are not guaranteed to find a global minimum. In order to improve the bound they provide for $\eta^*(\rho_{AB},N,k)$, multiple different initial points can be tested. They also provide a candidate for the optimal set of $N$ $k$-outcome measurements in the given scenario, the one that generates the most robust assemblage when locally performed on $\rho_{AB}$.
\\
\textit{See-saw algorithm}. The seesaw algorithm is an iterative method for solving some nonlinear optimization problems that has found many applications in quantum information theory. In Refs. \cite{CounterPeres,DisprovePeres,RevSDP} seesaw algorithms are used as methods of measurement optimization that are here adapted to approach our problem.
Our seesaw iterates two SDPs. The first one is the dual formulation of SDP (\ref{wnrsdp}):
\begin{subequations}
\begin{align}
\begin{split}\label{seesaw1}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{M_{a|x}\} \\
\min_{\{F_{a|x}\}} &\hspace{0.2cm} 1 - \sum_{a,x} \text{\normalfont Tr}(F_{a|x}\sigma_{a|x}) \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}) ,\ \forall\, a,x \\
& \hspace{0.2cm} 1 - \sum_{a,x} \text{\normalfont Tr} (F_{a|x}\sigma_{a|x}) + \frac{1}{d} \sum_{a,x} \text{\normalfont Tr} (F_{a|x})\text{\normalfont Tr}(\sigma_{a|x}) = 0 \\
& \hspace{0.2cm} \sum_{a,x} D_\lambda(a|x)F_{a|x} \leq 0, \ \forall\,\lambda.
\end{split}
\end{align}
This SDP returns the coefficients $\{F_{a|x}\}$ of a steering inequality of the form $\sum_{a,x}\text{\normalfont Tr}(F_{a|x}\sigma_{a|x})\geq0$. The value obtained by the assemblage $\{\sigma_{a|x}\}$, which is generated by the input state $\rho_{AB}$ and set of measurements $\{M_{a|x}\}$, for the left hand side of this inequality is precisely $1-\eta(\{\sigma_{a|x}\})$. This is due to the fact that primal and dual problems satisfy strong duality. As part of the seesaw, this SDP starts by taking a randomly chosen set of $N$ $k$-outcome measurements and the quantum state whose steerability one wishes to characterize.
The second SDP of the seesaw is
\begin{align}
\begin{split}\label{seesaw2}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{F_{a|x}\} \\
\max_{\{M_{a|x}\}} &\hspace{0.2cm} \sum_{a,x} \text{\normalfont Tr}(F_{a|x}\sigma_{a|x}) \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}) ,\ \forall\, a,x \\
&\hspace{0.2cm} M_{a|x}\geq 0, \ \forall\, a,x \\
& \hspace{0.2cm} \sum_a M_{a|x} = \mathbb{1}, \ \forall\,x.
\end{split}
\end{align}
\end{subequations}
This SDP takes the coefficients $\{F_{a|x}\}$ of the steering inequality that were outputted by the first SDP, (\ref{seesaw1}), as input and, for the same quantum state $\rho_{AB}$, finds the set of POVMs $\{M_{a|x}\}$ that generates the assemblage that maximally violates this inequality. The measurement set that is the output of this SDP, (\ref{seesaw2}), will be the input of the first SDP, (\ref{seesaw1}), in the next round of the iteration. When performed locally on the fixed quantum state, it will necessarily generate an assemblage that has the same or a lower critical visibility than the measurement set from the previous round. When some convergence condition is satisfied (e.g., the diference between the solutions of SDP (\ref{seesaw1}), in two subsequent rounds is less than a certain value) the iteration is halted. The final value for $\eta$ found by the seesaw is an upper bound for $\eta^*(\rho_{AB},N,k)$ of the input state and the set of measurements found by the seesaw is a candidate for the optimal set of $N$ general measurements with $k$ outcomes for steering the state $\rho_{AB}$.
This is also a heuristic method, hence, by itself, it does not prove that the obtained bound is tight. However, it is possible to improve the result by testing multiple different initial points. Contrary to the search algorithm, which allows for constraints on the structure of the POVMs, this method optimizes over all possible sets of $N$ $k$-outcome general POVMs. Our calculations have shown that even though the seesaw algorithm does not have this extra feature, when the interest is in optimizing over general POVMs, it is more effective in doing so than the search algorithm (in the sense that the seesaw demands computational times that are orders of magnitude smaller than the search algorithm for the same number of measurements and outcomes). In all cases tested, for the same state and scenario the solutions of both methods coincide.
\subsection{Lower bounds for $\eta^*(\rho_{AB},N,k)$}\label{lower}
\textit{Outer polytope approximation.} Consider the set $\set{A}$ of all assemblages that can be generated by performing $N$ local measurements of $k$ outcomes on a fixed quantum state $\rho_{AB}$. This set is convex but not a polytope. In order to guarantee that all assemblages in $\set{A}$ admit an LHS model it is sufficient to guarantee that this holds for all of the extremal assemblages of the set. However, since there is an \textit{infinite number} of extremal assemblages in this set (each one corresponding to an extremal set of $N$ $k$-outcome measurements) it is not viable to test each and every one of them.
The method we propose to overcome this problem is based on the techniques presented in Ref. \cite{SuperPOVM}, where the authors approximate the set of quantum measurements by outer polytopes. The idea is to construct an external polytope $\Delta$ that contains $\set{A}$, such that every assemblage in $\set{A}$ can be expressed as a convex combination of the \textit{finitely many} extremal points of $\Delta$. We call the extremal points of $\Delta$ quasiassemblages (they are nonpositive operators that sum to a reduced state $\rho_B$). The way we generate these quasi-assemblages is by applying well-chosen quasi-POVMs \cite{SuperPOVM} (nonpositive operators that sum to the identity) to a fixed quantum state. One can calculate the white-noise robustness of each quasi-assemblage in $\Delta$ using SDP (\ref{wnrsdp}), and the lowest value among them will be a lower bound for $\eta^*(\rho_{AB}, N,k)$. The SDP will return LHS models for the quasi-assemblages which can be used to construct, by simple convex combination, LHS models for all assemblages in the depolarized set.
We now detail the construction of these polytopes for the case where the dimension of Alice's system is $d=2$. A generalization to higher dimensions follows analogously, and we refer to Ref. \cite{SuperPOVM} for more details. In this case, any measurement operator $M$ can be written as $M =\alpha\mathbb{1}+\vec{v}\cdot\vec{\sigma}$, where $\vec{v}$ is a three-dimensional real vector and $\vec{\sigma}$ is the vector of Pauli matrices. By checking the eigenvalues we see that $M\geq0$ if and only if $||\vec{v}||\leq\alpha$, where $||\cdot||$ is the Euclidian norm, which is equivalent to saying that $\vec{v}$ is contained in a real sphere of radius $\alpha\geq 0$. This allows one to represent each measurement operator as a vector in a rescaled Bloch sphere of radius $\alpha$. In order to approximate the set of all POVMs in $d=2$ it is sufficient to approximate the Bloch sphere by an outer polyhedron, which is a simple task in $\mathbb{R}^3$ (see Fig. \ref{blochpolytope}).
\begin{figure}
\caption{\small Example of an approximation of the Bloch sphere by an outer polytope.}
\label{blochpolytope}
\end{figure}
Since the extremal points of the polytope are outside the Bloch sphere, i.e., $||\vec{v}||>\alpha$, they violate the positivity condition for operators in a two-dimensional Hilbert space. Hence, the extremal points of the polytope do not correspond to positive semidefinite operators. They are represented by a vector $v$ such that $v\cdot w_i\leq\alpha$, for some finite set $\{w_i\}$ of vectors defining facets of a polytope that contains the sphere of radius $\alpha$. Accordingly, a quasi-POVM is a set of these nonpositive operators that sum up to the identity. All sets of $N$ quasi-POVMs that can be constructed from the extremal points of the polytope that approximates the Bloch sphere are then locally performed on $\rho_{AB}$ to obtain the quasi-assemblages that define the polytope $\Delta$ that approximates the set $\set{A}$.
The lower bound provided by this method can be improved by increasing the number of tangency points of the outer polytope on the sphere. Contrary to the search and seesaw algorithms, the outer polytope method converges to the exact value of $\eta^*(\rho_{AB},N,k)$ with probability $p=1$ in the limit of an infinite number of generic extremal points. Hence, the bound can be improved as much as necessary, up to available computational resources. | 3,507 | 18,845 | en |
train | 0.28.3 | \subsection{Lower bounds for $\eta^*(\rho_{AB},N,k)$}\label{lower}
\textit{Outer polytope approximation.} Consider the set $\set{A}$ of all assemblages that can be generated by performing $N$ local measurements of $k$ outcomes on a fixed quantum state $\rho_{AB}$. This set is convex but not a polytope. In order to guarantee that all assemblages in $\set{A}$ admit an LHS model it is sufficient to guarantee that this holds for all of the extremal assemblages of the set. However, since there is an \textit{infinite number} of extremal assemblages in this set (each one corresponding to an extremal set of $N$ $k$-outcome measurements) it is not viable to test each and every one of them.
The method we propose to overcome this problem is based on the techniques presented in Ref. \cite{SuperPOVM}, where the authors approximate the set of quantum measurements by outer polytopes. The idea is to construct an external polytope $\Delta$ that contains $\set{A}$, such that every assemblage in $\set{A}$ can be expressed as a convex combination of the \textit{finitely many} extremal points of $\Delta$. We call the extremal points of $\Delta$ quasiassemblages (they are nonpositive operators that sum to a reduced state $\rho_B$). The way we generate these quasi-assemblages is by applying well-chosen quasi-POVMs \cite{SuperPOVM} (nonpositive operators that sum to the identity) to a fixed quantum state. One can calculate the white-noise robustness of each quasi-assemblage in $\Delta$ using SDP (\ref{wnrsdp}), and the lowest value among them will be a lower bound for $\eta^*(\rho_{AB}, N,k)$. The SDP will return LHS models for the quasi-assemblages which can be used to construct, by simple convex combination, LHS models for all assemblages in the depolarized set.
We now detail the construction of these polytopes for the case where the dimension of Alice's system is $d=2$. A generalization to higher dimensions follows analogously, and we refer to Ref. \cite{SuperPOVM} for more details. In this case, any measurement operator $M$ can be written as $M =\alpha\mathbb{1}+\vec{v}\cdot\vec{\sigma}$, where $\vec{v}$ is a three-dimensional real vector and $\vec{\sigma}$ is the vector of Pauli matrices. By checking the eigenvalues we see that $M\geq0$ if and only if $||\vec{v}||\leq\alpha$, where $||\cdot||$ is the Euclidian norm, which is equivalent to saying that $\vec{v}$ is contained in a real sphere of radius $\alpha\geq 0$. This allows one to represent each measurement operator as a vector in a rescaled Bloch sphere of radius $\alpha$. In order to approximate the set of all POVMs in $d=2$ it is sufficient to approximate the Bloch sphere by an outer polyhedron, which is a simple task in $\mathbb{R}^3$ (see Fig. \ref{blochpolytope}).
\begin{figure}
\caption{\small Example of an approximation of the Bloch sphere by an outer polytope.}
\label{blochpolytope}
\end{figure}
Since the extremal points of the polytope are outside the Bloch sphere, i.e., $||\vec{v}||>\alpha$, they violate the positivity condition for operators in a two-dimensional Hilbert space. Hence, the extremal points of the polytope do not correspond to positive semidefinite operators. They are represented by a vector $v$ such that $v\cdot w_i\leq\alpha$, for some finite set $\{w_i\}$ of vectors defining facets of a polytope that contains the sphere of radius $\alpha$. Accordingly, a quasi-POVM is a set of these nonpositive operators that sum up to the identity. All sets of $N$ quasi-POVMs that can be constructed from the extremal points of the polytope that approximates the Bloch sphere are then locally performed on $\rho_{AB}$ to obtain the quasi-assemblages that define the polytope $\Delta$ that approximates the set $\set{A}$.
The lower bound provided by this method can be improved by increasing the number of tangency points of the outer polytope on the sphere. Contrary to the search and seesaw algorithms, the outer polytope method converges to the exact value of $\eta^*(\rho_{AB},N,k)$ with probability $p=1$ in the limit of an infinite number of generic extremal points. Hence, the bound can be improved as much as necessary, up to available computational resources.
\subsection{Brief discussion of the methods}\label{methoddiscussion}
\textit{Different quantifiers of steering and joint measurability.} We start our discussion by remarking that although the presented methods were based on the white-noise robustness of steering, they can be easily adapted to estimate other quantifiers of steering that can be calculated by an SDP for fixed state and measurements. Some examples are the steering weight \cite{SteeringWeight} and the generalized robustness of steering \cite{Piani}. Also, given the strong connection between joint measurability and steering discussed in Sec. \ref{JM=STE}, analogues of all these steering quantifiers also exist for joint measurability \cite{QuantifiersDani} and our methods can be used to obtain upper and lower bounds for these quantities as well.
\textit{Convergence.} As discussed in Sec. \ref{lower}, the method to calculate lower bounds for $\eta^*(\rho_{AB},N,k)$ is constituted by a sequence of algorithms that converges to the precise value in the limit of infinite extremal points. The cumbersome feature is that the precise value cannot be attained within a finite number of steps. On the other hand, the upper-bound methods consist of heuristic optimization algorithms that may return the exact critical visibility, but there is no guarantee of that. We note that although we did not present a sequence of converging algorithms for calculating upper bounds for $\eta^*(\rho_{AB},N,k)$, one can be constructed by simply testing every possible combination of measurements, possibly with the assistance of polytopes that approximate the set of assemblages from the inside. Since the set of measurements is convex, it can be approximated by a converging sequence of polytopes, guaranteeing the existence of this sequence of algorithms \cite{siberian}. The drawback of this ``brute force'' converging method is that it may take an impractical amount of time to find useful bounds, which is not the case for the heuristic upper-bounds methods discussed in Sec. \ref{upper}.
\textit{Lower bounds for a finite vs. an infinite number of measurements.} In Refs. \cite{LHScavalcanti,LHShirsch}, the authors have presented a method for constructing LHS models for quantum states when all possible measurements (hence an infinite number) are considered. Here, we address a similar question, but in cases where a finite number of measurements is considered. Perhaps surprisingly, our algorithm suggests that constructing local hidden state models for only a finite number of measurements is considerably harder than for an infinite number of measurements. For instance, calculating (good) lower bounds for the critical visibility of two-qubit Werner states subjected to five dichotomic measurements was a very computationally challenging task. Nonetheless, when all possible measurements are considered, the numerical methods of Refs. \cite{LHScavalcanti,LHShirsch} can find good lower bounds in a reasonably small time.
\textit{Numerical stability of the seesaw method.} When implementing the seesaw method with the visibility parametrization described in Sec. \ref{Quantify}, we faced some numerical instability. To overcome this problem, the parametrization
\begin{equation}
\frac{\sigma_{a|x} + t \text{\normalfont Tr}(\sigma_{a|x})\frac{1}{d}}{1+t} \in LHS
\end{equation}
was used instead. The SDPs were then rewritten as a minimization over the parameter $t$ with the correspondence $\eta^*=\frac{1}{1+t^*}$, where the superscript $^*$ denotes the optimal value. Although the interpretation of the visibility parametrization is more straightforward due to its relation with the depolarazing map, the formulation of the problem with the $t$ parameter is equivalent. The numerical stability was also improved by avoiding redundant constraints on normalization and nonsignaling conditions.
We also used the seesaw method to calculate upper bounds for the generalized robustness of steering \cite{Piani} of a quantum state. The seesaw for this quantifier was shown to be more numerically stable than the one for white-noise robustness. As a consequence, the generalized robustness seesaw was used to approach the scenarios with the largest number of parameters in this work. | 2,057 | 18,845 | en |
train | 0.28.4 | \section{Results}
We now present the results we obtained by applying the machinery developed in the last section to some specific quantum states. In order to tackle the steering and the joint measurability problem simultaneously, we concentrate on isotropic states in our examples. Also, two-qubit isotropic states can be mapped into two-qubit Werner states \cite{WernerModel} via a local unitary transformation, which always preserves the steerability \cite{MTGeneralMeas}. For this reason, we present our two-qubit results in terms of Werner states, which in this case are given by
\begin{equation}
(\mathbb{1}\otimes\Lambda_\eta)(\ketbra{\psi^-}{\psi^-})=\eta\ketbra{\psi^-}{\psi^-}+(1-\eta)\frac{\mathbb{1}}{4},
\end{equation}
where $\ket{\psi^-}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$ is the singlet state.
To simplify notation we refer to the critical visibility of these two-qubit states as simply $\eta^*(N,k)\coloneqq\eta^*(2,N,k)$.
\subsection{Planar qubit projective measurements}\label{planar}
We start with a simple family of qubit measurements, the planar projective measurements. These are qubit projective measurements whose Bloch vectors are confined to the same plane. The reasons for studying this kind of measurement include its simple experimental implementation \cite{WisemanPlanar} and the low computational cost required to optimize over these measurements, compared to more general ones.
Initially, we use the search algorithm with the constraint that all measurement vectors are coplanar to calculate upper bounds for the critical visibility $\eta^*(N,2)$ of two-qubit Werner states. Calculations were performed for sets of $N\in\{2,...,15\}$ planar projective measurements. The results are presented in Fig. \ref{graphprojective} and Table \ref{tablenum}.
For all trials performed with multiple different initial points, the result for both the objective function--the parameter $\eta$--and the optimization variables--the angles between the Bloch vectors of the measurements--were the same for all values of $N$ tested. In all cases, the optimal set of measurements found by the algorithm is the one in which the Bloch vectors of all measurements are equally spaced on a plane, i.e., each Bloch vector is separated from its next neighbors by an angle of $\frac{\pi}{N}$, as represented for the cases of $N\in\{2,...,5\}$ in Fig. \ref{figoptsetplan}.
\begin{figure}
\caption{\small Plot of the upper bounds of the critical visibility of two-qubit Werner states subjected to $N$ regular tetrahedron and regular trine measurements and the upper and lower bounds for $N$ planar projective and general projective measurements. Dotted black lines correspond to the values $\eta=\frac{1}
\label{graphprojective}
\end{figure}
Next, we calculated lower bounds for $\eta^*(N,2)$ in the restricted scenario of planar projective measurements using the method of outer polytope approximation. Results are reported for the cases of $N\in\{2,\ldots,5\}$ planar projective measurements also in Fig. \ref{graphprojective} and Table \ref{tablenum}. The lower bound for $\eta^*(N,2)$ found by the outer polytope approximation method matches the upper bound found by the search algorithm up to three or four decimal places for all cases tested. We consider this to be enough evidence to claim that for the cases of $N\in\{2,\ldots,5\}$ planar projective measurements, the optimal set of measurements for steering two-qubit Werner states is the set of equally spaced measurements. This is equivalent to stating that the most incompatible set of $N\in\{2,\ldots,5\}$ planar projective qubit measurements is the set of equally spaced measurements. We also conjecture this result to be valid for any number of planar projective measurements. The values we calculated match the analytical results for the incompatibility of equally spaced planar projective qubit measurements presented in Refs. \cite{WisemanPlanar,Siegen}.
\begin{figure}
\caption{\small Optimal set of $N\in\{2,\ldots,5\}
\label{figoptsetplan}
\end{figure}
\subsection{General qubit projective measurements}\label{spherical}
Since the optimal sets of measurements for our problem in the case of planar projective measurements appear to be the sets of equally spaced measurements on a plane, we hypothesize that the optimal sets of general qubit projective measurements correspond to some notion of equally spaced points on a sphere. Unfortunately, contrarily to the equivalent problem on a circumference, the problem of equally distributing points on a sphere is not trivial and many different sets of points can be defined using different notions of distance. This problem is particularly difficult in the regime of few points. For this work we chose the equally spaced notion of the Thomson problem \cite{Thomson} and the Fibonacci problem \cite{Fibonacci}. The former, for the particular cases of $N\in\{2,3,4,6,10\}$ projective measurements corresponding to $4,6,8,12,$ and $20$ vertices, is equivalent to the Platonic solids. The results for the critical visibility of two-qubit Werner states subjected to sets of $N\in\{2,\ldots,18\}$ local measurements constructed from these two notions of equal spacing are listed in Table \ref{tablenum}.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c | c c |}
\multicolumn{7}{c}{Projective qubit measurements} \\
\hline
\multicolumn{1}{|c}{} & \multicolumn{2}{c }{Gen. opt.} & \multicolumn{2}{c |}{Planar opt.} & \multicolumn{2}{c |}{Fixed sets} \\
\hline \hline
$N$ & Upper & Lower & Upper & Lower & Thomson & Fibonnaci \\
\hline
$\quad$ $2$ $\quad$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7102$\\
$3$ & $0.5774$ & $0.5755$ & $0.6667$ & $0.6667$ & $0.5774$ & $0.6981$ \\
$4$ & $0.5547$ & $0.5437$ & $0.6533$ & $0.6532$ & $0.5774$ & $0.6114$ \\
$5$ & $0.5422$ & $0.5283$ & $0.6472$ & $0.6470$ & $0.5513$ & $0.5653$ \\
$6$ & $0.5270$ & $$ & $0.6440$ & $$ & $0.5393$ & $0.5561$ \\
$7$ & $0.5234$ & $$ & $0.6420$ & $$ & $0.5234$ & $0.5533$ \\
$8$ & $0.5202$ & $$ & $0.6407$ & $$ & $0.5250$ & $0.5508$ \\
$9$ & $0.5149$ & $$ & $0.6399$ & $$ & $0.5209$ & $0.5359$ \\
$10$ & $0.5144$ & $$ & $0.6392$ & $$ & $0.5191$ & $ 0.5302$ \\
$11$ & $0.5132$ & $$ & $0.6388$ & $$ & $0.5148$ & $0.5274$ \\
$12$ & $0.5117$ & $$ & $0.6384$ & $$ & $0.5152$ & $0.5261$ \\
$13$ & $0.5105$ & $$ & $0.6382$ & $$ & $0.5126$ & $0.5220$ \\
$14$ & $$ & $$ & $ 0.6380$ & $$ & $0.5114$ & $ 0.5180$ \\
$15$ & $$ & $$ & $0.6378$ & $$ & $ 0.5107$ & $0.5158$ \\
$16$ & $$ & $$ & $$ & $$ & $0.5106$ & $0.5158$ \\
$17$ & $$ & $$ & $$ & $$ & $0.5086$ & $0.5150$ \\
$18$ & $$ & $$ & $$ & $$ & $0.5079$ & $0.5136$ \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for the critical visibility of two-qubit Werner states subjected to $N$ projective measurements.}
\label{tablenum}
\end{table}
To test whether these sets of measurements are indeed optimal, we once again use the search algorithm, now with the only restriction that the measurement operators correspond to projectors. We report upper bounds for the value of $\eta^*(N,2)$ in scenarios of $N\in\{2,\ldots,13\}$ general projective measurements. In all cases the search algorithm was able to improve the bound provided by both the Thomson and the Fibonacci measurements (see Table \ref{tablenum}), proving that they are actually not optimal. The best upper bounds are plotted in Fig. \ref{graphprojective} and the Bloch vectors of the measurement elements that form the best candidate for the optimal set of measurements in the cases of $N\in\{2,\ldots,6\}$ projective measurements are plotted in Fig. \ref{figoptsetsph}.
\begin{figure}
\caption{\small Candidates for the optimal set of $N\in\{2,\ldots,6\}
\label{figoptsetsph}
\end{figure}
For these measurements, the vectors are distributed in a particular way: for $2$ and $3$ measurements, we have sets of orthogonal vectors; for $4$ measurements we have $3$ coplanar and equally distributed vectors and $1$ vector orthogonal to the other $3$; for $5$ and $6$ measurements, the structure of $3$ coplanar equally spaced vectors is maintained and the other vectors are agglomerated around the poles of the sphere with the same $z$-projection. For the cases of $7$ or more measurements, this apparent symmetry is no longer necessarily respected.
Using the outer polytope approximation we calculated the lower bounds for the cases of $N\in\{2,\ldots,5\}$ projective measurements that can be seen on Fig. \ref{graphprojective} and Table \ref{tablenum}. In this case the gap between upper and lower bounds for the general projective case is larger than for the planar projective case. This is due to the increase in the number of parameters in the former case as compared to the latter. However, due to the convergence properties of our outer polytope method, as discussed on Secs. \ref{lower} and \ref{methoddiscussion}, these lower bounds can be improved beyond what is the scope of this work. | 2,886 | 18,845 | en |
train | 0.28.5 | \subsection{General POVM relevance for qubits}\label{povm}
An old standing question in nonlocality is to understand when general POVMs are useful to reveal this property in a given quantum state \cite{WernerModel,Barrett,WernerPOVM,nguyen16}. It is well known that constructing local hidden variable (LHV) and LHS models for general POVMs is considerably harder than constructing these models for projective measurements \cite{WernerModel,Barrett,WernerPOVM,nguyen16}. Moreover, it is not known whether nonprojective measurements are more useful than projective ones to demonstrate EPR steering or Bell nonlocality. For some particular fixed (nontight) Bell inequalities, it is known that general POVMs can lead to a larger Bell violation than projective measurements \cite{RelevantPOVM}, but the existence of a quantum state that has a LHV/LHS model for projective measurements but displays Bell nonlocality/EPR steering when general POVMs are considered is still an open question.
We have applied our seesaw method to two-qubit Werner states where the noncharacterized party has access to $N\in\{2,\ldots,7\}$ general POVMs with $2,3,$ and $4$ outcomes. We recall that for the case of $2$ outputs, nonprojective POVMs can never be useful for nonlocality, since they can always be written as convex combinations of projective measurements \cite{ExtremalPOVM}. Also, qubit POVMs with more than $4$ outcomes are never extremal \cite{ExtremalPOVM}, hence these measurements could never lead to better bounds for the critical visibility. For this reason we now define the quantity $\eta^*(N)\coloneqq\eta^*(N,d^2=4)$, the critical visibility of two-qubit Werner states when subjected to $N$ POVMs of an arbitrary number of outcomes.
In addition to using the seesaw method to explore general POVMs, we have applied the search algorithm to the specific case where Alice is required to perform symmetric $3$- and $4$-outcome POVMs on her side of a maximally entangled two-qubit state. In the $4$-outcome case, we have fixed all measurements to be SIC-POVMs \cite{renes04}, which are extremal measurements whose Bloch vectors correspond to the vertices of a regular tetrahedron. In the $3$ outcomes case, the chosen POVM was the symmetric extremal measurement whose Bloch vectors correspond to the ``Mercedes-Benz star'', also called the regular trine \cite{Trine} (see Fig. \ref{optpovm}). These particular symmetric nonprojective measurements are known to be useful in tasks such as tomography \cite{rehacek04,SICPOVMTomography} and cryptography \cite{fuchs03}, hence they are interesting examples of extremal nonprojective qubit POVMs \cite{ExtremalPOVM}.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c |}
\multicolumn{4}{c}{Symmetric qubit POVMs} \\
\hline
$N$ & Proj. ($k=2$) & Trine ($k=3$) & Tetra. ($k=4$) \\
\hline
$\quad$ $2$ $\quad$ & $\quad$ $0.7071$ $\quad$ & $\quad$ $0.7739$ $\quad$ & $\quad$ $0.8165$ $\quad$\\
$3$ & $0.5774$ & $0.7202$ & $0.7829$ \\
$4$ & $0.5547$ & $0.6917$ & $0.7716$ \\
$5$ & $0.5422$ & $0.6791$ & $0.7653$ \\
$6$ & $0.5270$ & $0.6690$ & $0.7617$ \\
$7$ & $0.5234$ & $0.6656$ & $0.7605$ \\
$8$ & $0.5202$ & $0.6647$ & -- \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for upper bounds of the critical visibility of two-qubit Werner states subjected to $N$ extremal symmetric POVMs.}
\label{tablesic}
\end{table}
Our results for symmetric qubit POVMs are plotted in Fig. \ref{graphprojective} and listed in Table \ref{tablesic} for the cases of $N\in\{2,\ldots,8\}$, including the results for projective measurements, which are symmetric $2$-outcome POVMs, for the sake of comparison. In the case of $N=2$, the optimal set of regular trine and regular tetrahedron POVMs is plotted in Fig. \ref{optpovm}. It is easy to see that under none of the analyzed scenarios were the symmetric nonprojective POVMs able to show more steering than the projective measurements. In fact, the bounds for symmetric $3$- and $4$-outcome qubit POVMs are considerably worse than for projective qubit measurements.
\begin{figure}
\caption{\small Candidates for the optimal set of $N=2$ regular trine (left; $k=3$) and regular tetrahedron (right; $k=4$) symmetric qubit POVMs for steering two-qubit Werner states.}
\label{optpovm}
\end{figure}
As for the optimization over general $3$- and $4$-outcome qubit POVMs, we could \textit{not} find any set of $N$ general POVMs that are able to overperform projective ones. For $N=2$ and $3$ and $k=4$, the seesaw algorithm ran $10^5$ times, each time with a different initial point; for $N=4$ and $k=3$, the seesaw ran $4\times10^4$ times, and for $k=4$, $3\times10^4$ times; for $N=5$, $6$, and $7$, and $k=3$, it ran $2\times10^4$, $2\times10^3$, and $200$ times, respectively. Without exception, our algorithm recovered the bound for $\eta^*(N)$ obtained by the optimization over projective measurements using general POVMs, usually by nulling two measurement outcomes and ``simulating'' a projective measurement. However, it was never able to surpass it.
Strictly speaking, the results presented in this section are only upper bounds for the critical visibility $\eta^*(N)$. Nevertheless, given the small number of parameters in the two-qubit scenario and the number of times we have ran our heuristic method, we believe that these results are strong evidence that general POVMs are not useful to reveal EPR-steering in two-qubit Werner states.
\subsection{Higher dimension states and measurements}\label{higherd}
We now explore the generality of our seesaw method on quantum systems of dimension $d>2$ by calculating bounds for the critical visibility $\eta^*(d,N,k)$ of higher dimension maximally entangled states. Let us start with the simple case where these states are subjected to only $2$ local general measurements. These calculations are reported for states of dimensions $d\in\{2,\ldots,6\}$ in Table \ref{tablen2}. We note that by increasing the number of outcomes in the measurements from $k=2$ up to $k=d$ the bounds for $\eta^*(d,2,k)$ are significantly improved and the optimal sets of measurements are always composed by projective measurements--even though most of these scenarios allow extremal nonprojective POVMs. However, once the number of outcomes achieves $k=d+1$ the bound for $\eta^*(d,2,k)$ provided by the seesaw method ceases to decrease and it seems that increasing the number of outputs beyond this point does not improve the results. Since there only exist projective measurements with up to $k=d$ outcomes, this result is evidence that allowing POVMs more general than projective measurements does not increase the robustness of the steerability of isotropic states. Following the connection between the steerability of these states and joint measurability, this is also evidence that sets of $2$ general qudit POVMs cannot be more incompatible than sets of $2$ projective qudit measurements.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c c |}
\multicolumn{6}{c}{$N=2$} \\
\hline
$\quad$ $k$ $\quad$& $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.7000$ & $0.6901$ & $0.6812$ & $0.6736$ \\
$3$ & $0.7071$ & $0.6794$ & $0.6722$ & $0.6621$ & $0.6527$ \\
$4$ & $$ & $0.6794$ & $0.6665$ & $0.6544$ & $0.6448$ \\
$5$ & $$ & $$ & $0.6665$ & $0.6483$ & $0.6429$ \\
$6$ & $$ & $$ & $$ & $0.6483$ & $0.6390$ \\
$7$ & $$ & $$ & $$ & $$ & 0.6390 \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for upper bounds of the critical visibility of $d$-dimensional isotropic states subjected to $2$ general POVMs of $k\in\{2,\ldots,d+1\}$ outcomes.}
\label{tablen2}
\end{table}
Since the scenario where the uncharacterized party is allowed to perform only $2$ measurements is very particular, we performed the same calculations reported above for $d$-dimensional isotropic states allowing scenarios with $3$ and $4$ general measurements with outcomes up to $d+1$ as well. In these broader scenarios, we also calculated upper bounds for the critical visibility $\eta^*(d,N,k)$ of the isotropic states. However, since the number of parameters increases too rapidly (exponentially on the number of measurements), the seesaw method presented some numerical instability, and for this reason we are not able to reach any conclusions about the relevance of general POVMs in these scenarios.
\subsection{Mutually unbiased bases}\label{MUB}
A set of MUBs consists of 2 or more orthonormal bases $\{\ket{a_x}\}_a$ in a $d$-dimensional Hilbert space that satisfy
\begin{equation}
|\braket{a_x}{b_y}|^2=\frac{1}{d}, \quad \forall\, a,b \in\{1,\ldots,d\}, \ x\neq y,
\end{equation}
for all bases $x,y$ \cite{durt10}. A set of MUBs is called complete if for a Hilbert space of dimension $d$ there exists $d+1$ MUBs. These bases can be used to construct sets of mutually unbiased projective measurements with a high level of symmetry, and for this reason one might think they would be good candidates for the optimal set of measurements for measurement incompatibility and for EPR steering with a maximally entangled state.
We have calculated the critical visibility of isotropic states of dimension $d\in\{2,\ldots,6\}$ when subjected to local MUB measurements using SDP (\ref{wnrsdp}) and listed the results in Table \ref{tablemub}. These exact values calculated by our SDP (\ref{wnrsdp}), show significant improvement over the analytical bounds obtained in Refs. \cite{marciniak15,hgsieh16} for steering with MUBs and maximally entangled states. Next, we used the seesaw method to calculate upper bounds for $\eta^*(d,N,d)$ of the isotropic states when locally subjected to sets of general POVMs with $d$ outcomes for some number of measurements $N$ where MUB measurements are known to exist. Perhaps surprisingly, in many cases we found sets of measurements with greater or equal robustness, showing that MUBs are not necessarily the best choice of measurements to reveal quantum steering nor are they the most incompatible ones. The results are listed in Table \ref{tablemub}. The optimal measurements found by the seesaw method are all projective measurements in these cases as well. We remark that in Refs. \cite{carmeli11,heinosaari13}, the authors have computed (analytically) the required visibility $\eta^*(d,2,d)$ for any pair of $d$-dimensional MUB measurements to be jointly performed; here we have shown that there exist pairs of measurements that are even more incompatible than mutually unbiased ones. However, in scenarios where there exist complete sets of MUB measurements, for dimensions $2$, $3$, and $4$, we were not able to find measurements more resistant to white noise and better for steering isotropic states than the MUB ones, which is evidence that they may be optimal for this task.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c c |}
\multicolumn{6}{c}{MUBs} \\
\hline
$\quad$ $N$ $\quad$ & $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.6830$ & $0.6667$ & $0.6545$ & $0.6449$ \\
$3$ & $0.5774$ & $0.5686$ & $0.5469$ & $0.5393$ & $0.5204$ \\
$4$ & $$ & $0.4818$ & $0.5000$ & $0.4615$ & $$ \\
$5$ & $$ & $$ & $0.4309$ & $0.4179$ & $$ \\
$6$ & $$ & $$ & $$ & $0.3863$ & $$ \\
\hline
\multicolumn{6}{c}{} \\
\multicolumn{6}{c}{General $d$-outcome POVMs} \\
\hline
$N$& $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.6794$ & $0.6665$ & $0.6483$ & $0.6395$ \\
$3$ & $0.5774$ & $0.5572$ & $0.5412$ & $0.5266$ & $0.5139$ \\
$4$ & $$ & $0.4818$ & $0.4797$ & $0.4615$ & $$ \\
$5$ & $$ & $$ & $0.4309$ & -- & $$ \\
$6$ & $$ & $$ & $$ & -- & $$ \\
\hline
\end{tabular}
}
\end{center}
\caption{Comparison between the exact critical visibility of isotropic states in dimension $d$ subject to local mutually unbiased measurements and the upper bound of the same states when optimizing over general POVMs with $k=d$.}
\label{tablemub}
\end{table} | 3,983 | 18,845 | en |
train | 0.28.6 | \section{Discussion}
We have used three methods for investigating EPR steering and joint measurability under restrictive measurement scenarios and discussed the applicability of each one. Using white-noise robustness as a quantifier, we have presented two heuristic methods for calculating the critical visibility of quantum states subjected to a finite number of measurements and one converging method for lower-bounding the same quantity. Our methods can be easily adapted to other steering and joint measurability quantifiers.
For two-qubit Werner states, we showed that the best sets of $N\in\{2,\ldots,5\}$ planar projective measurements are equally spaced measurements and conjecture this result to be valid for all $N\in\mathbb{N}$. Our upper bounds for the critical visibility of two-qubit Werner states subjected to planar projective measurements match the analytical expressions derived in Refs. \cite{WisemanPlanar,Siegen} for equally spaced measurements. We proved that intuitive notions of equally spaced measurements in the Bloch sphere, like the vertices of Platonic solids, do not correspond to the best measurements to show steering with two-qubit Werner states, nor are they the most incompatible sets of measurements. We showed that symmetric $3$- and $4$-outcome qubit POVMs are not optimal for steering two-qubit Werner states as well. Upper bounds for the critical visibility of two-qubit Werner states subjected to $N\in\{2,\ldots,18\}$ general measurements were calculated. We provided strong numerical evidence that general POVMs are not more suitable for steering two-qubit Werner states than projective measurements, and suggested candidates for the optimal sets of $N\in\{2,\ldots,6\}$ qubit measurements that are projective and follow a nonintuitive pattern.
Our results for higher dimension isotropic states indicate that increasing the number of outcomes until $k=d$ improves the bound for the critical visibility of the state. However, increasing the number of outcomes beyond the value of the local dimension of the state does not seem to improve the bounds, which strengthens the idea that nonprojective POVMs are not relevant for steering. The candidates for optimal measurements in all higher dimension scenarios are projective measurements. Finally, we proved that many incomplete sets of MUB measurements are not optimal for steering and provided numerical evidence that complete sets of MUB measurements could be optimal for steering isotropic states.
Although we presented numerical evidence against the relevance of nonprojective POVMs for EPR steering, deciding if projective measurements are indeed optimal for steering in all scenarios and for all quantum states still remains an open question.
One future direction is to apply similar techniques for the study of Bell nonlocality. Although some simple adaptation of our methods can be used to tackle the analogous problem for Bell nonlocality, the number of parameters in the problem could make our algorithms impracticable even in simple scenarios.
All code written for this work is available in a repository \cite{Code}.
{\small{
}}
\end{document} | 689 | 18,845 | en |
train | 0.29.0 | \betaegin{document}
\deltaate{\tauoday}
\tauitle{Jet schemes of quasi-ordinary surface singularities}
\alphauthor{Helena Cobo}
\varepsilonmail{[email protected]}
\alphauthor{Hussein Mourtada}
\alphaddress{Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7, B\^atiment Sophie Germain, 75013 Paris, France.}
\varepsilonmail{[email protected]}
\betaegin{abstract}
In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity.
We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the
topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on $\mathbb A^3$), that computes the log canonical threshold
of the singularity embedded in $\mathbb A^3$. This provides us with pairs $X\sigmaubset\mathbb A^3$ whose log canonical thresholds are not contributed by monomial divisorial valuations.
Note that for a pair $C\sigmaubset\mathbb A^2$, where $C$ is a plane curve, the log canonical threshold is always contributed by a monomial divisorial valuation (in suitable coordinates of $\mathbb A^2$).
\varepsilonnd{abstract}
\sigmaubjclass[2010]{14E18,14J17.}
\keywords{Singularities, Jet schemes, Quasi-ordinary singularities, log canonical threshold.}
\title{Jet schemes of quasi-ordinary surface singularities}
\sigmaection{Introduction}
A quasi-ordinary singularity $(X,0)$ of dimension $d,$ comes with a finite projection $p:X\lambdaongrightarrow \mathbb{A}^d,$ whose discriminant is a normal crossing
divisor. These singularities appear in the Jungian approach to resolution of singularities (see \cite{PP-HJ}). We are interested in irreducible quasi-ordinary
hypersurfaces. Thanks to Abhyankar-Jung theorem, we know that a hypersurface of this type can be parametrized by a Puiseux series (i.e an element in
$\mathbb C[[x_1^{\pihirac{1}{n}},\lambdadots,x_d^{\pihirac{1}{n}}]]).$ Moreover, some special exponents (called the characteristic exponents) which belong to the support of
this series, are complete invariants of the topological type of the singularity (see \cite{Gau}). In particular, they determine invariants which come from
resolution of singularities, like the log canonical threshold or the Motivic zeta functions (\cite{BGG}, \cite{ACLM}, \cite{CoGPqo}). They also give
insights about the construction of a resolution of singularities (\cite{BMc0}, \cite{BMc}, \cite{GP4}, \cite{Vill}).
Our aim is to construct some comparable complete invariants for all type of singularities. We search for such invariants in the jet schemes. For $m\in \mathbb{N},$ the
$m$-jet scheme, denoted by $X_m$, is a scheme that parametrizes morphisms $\mbox{Spec }\mathbb C[t]/(t^{m+1})\lambdaongrightarrow X.$ Intuitively we can think of it as
parametrizing arcs in an ambient space, which have a large contact, depending on $m$, with $X.$ We know already that some invariants which come from resolution of singularities
are encoded in jet schemes (\cite{Mus}, \cite{EinMustata}).
We want to extract from the jet schemes information about the singularity, which can be expressed in terms of invariants of resolutions of singularities.
With this, our next goal is to construct a resolution of singularities by using invariants of jet schemes. For specific types of singularities, the knowledge of the irreducible components of the jet schemes $X_m$
of a singular variety X, together with some invariants of them, like dimension or embedding dimension, permits to determine deep invariants of the singularity of $X$: the topological type in the case of curves (see \cite{Hcur}),
and the analytical type in the case of normal toric surfaces (see \cite{HCRAS} and \cite{Htor}). Moreover, in the case of irreducible plane curves, the minimal embedded resolution can be constructed from the jet schemes (\cite{LMR}), and the same for rational double point singularities (\cite{MP}).
Notice that, understanding the structure of jet schemes for particular singularities, remains a difficult problem. These structures have been studied in \cite{Yuen}
and \cite{DoC} for determinantal varieties, in \cite{Hcur}
for plane curve singularities, \cite{HCRAS} and \cite{Htor} for normal toric surfaces, in \cite{Hrat} for rational double point surface singularities, and in
\cite{SS} for commuting matrix pairs schemes. In the toric case, no result is known for dimension bigger than two.
More general jet schemes can be defined, and there are relations with invariants of singularities (see \cite{Mus14}).
In this paper, we study jet schemes of a two-dimensional quasi-ordinary hypersurface singularity $X$. We will give a combinatorial description of the irreducible components of the set of $m$-jets with center
in the singular locus of $X$, in terms of the following invariants of the singularity: the lattices $N_0,N_1,\lambdadots,N_g,$ the minimal system of generators $\gammaamma_1,\lambdadots,\gammaamma_g$ of the semigroup $\Gamma$ of $X,$
and the numerical data attached to them, $n_1,\lambdadots,n_g$ and $e_1,\lambdadots,e_g$ (see Section \rhoef{secQO} for definitions). Given $h\in\mathbb C[x_1,\lambdadots,x_n]$, an algebraic variety $X$, and $p,m\in\mathbb Z_{>0}$ with $p\lambdaeq m$,
let $\mbox{Cont}_X^p(h)_m$ be the locally closed set defined as $\mbox{Cont}_X^p(h)_m=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(h\circ\gammaamma)=p\}$. Then we associate with any lattice point $\nu\in N_0$ the constructible set
\[D_m^\nu=(\mbox{Cont}_X^{\nu_1}(x_1)_m\cap\mbox{Cont}_X^{\nu_2}(x_2)_m)_{red},\]
and its closure $C_m^\nu=\mathbb overline{D_m^\nu}$. The sets $C_m^\nu$ are the candidates to be the irreducible components, but there are many inclusions among them. We study these inclusions by defining, on the lattices
associated with $X$, subtle relations which depend on the singular loci of the quasi-ordinary surfaces defined by the approximated roots.
For some components the relation is very easy, is given by the product ordering $\lambdaeq_p$, and we can prove that
\[\mbox{ if }\nu\lambdaeq_p\nu'\mbox{ then }C_m^{\nu'}\sigmaubset C_m^\nu.\]
For other components the relation is more complicated, we have that
\[\mbox{ if }\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}\mbox{ then }C_m^{\nu'}\sigmaubset C_m^\nu\]
where $j'(m,\nu)$ is the integer $j\in\{0,\lambdadots,g\}$ defined by
\[n_je_j\lambdaangle\nu,\gammaamma_j\rhoangle+e_j\lambdaeq m<n_{j+1}e_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+e_{j+1},\]
and $\sigma_{Reg,j'(m,\nu)}$ is defined to keep track of the singular locus of the $j'(m,\nu)$-th approximated root as follows.
Since the normalization of any quasi-ordinary singularity is a toric variety, let $\nu_j: Z_{\sigma,N_j}\lambdaongrightarrow V(f_j)$ be the normalization of $V(f_j)$ ($f_j$ being the $j$-th approximated root). Then $Z_{\sigma,N_j}\sigmaetminus\nu_j^{-1}(Sing(V(f_j)))$ is an open set of the toric variety $Z_{\sigma,N_j}$, it is a union of orbits, and we denote by $\sigma_{Reg,j}$ the fan formed by the faces of $\sigma$ corresponding to the orbits.
Then, with the minimal elements with respect to these relations we define a set $F_m\sigmaubset\mathbb Z^2$, and for any $\nu\in F_m,$ we have a component $C_m^\nu\sigmaubset X_m$. We prove that these are the irreducible components.
\betaegin{The}
For any $m\in\mathbb Z_{>0}$ we have that the space of $m$-jets of $X$ with center in the singular locus has the following decomposition into irreducible components
\[\pii_m^{-1}(X_{Sing})=\cup_{\nu\in F_m}C_m^\nu.\]
\varepsilonnd{The}
Hence, for
every $m \in \mathbb Z_{>0}$, we determine the irreducible components of $X_m$ with center in the singular locus of $X$. Moreover, we give a formula for the codimension of $C_m^\nu$ in Proposition \rhoef{Prop1}.
To prove the theorem we need to understand the geometry of the sets $D_m^\nu$. This is done by comparing their geometries with
the geometries of the corresponding subsets for the approximate roots. The following proposition describes this comparison.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in\sigma\cap N_j$ such that $n_1\cdots n_g\lambdaangle\nu,\gammaamma_1\rhoangle\lambdaeq m$, we have that
\[D_m^\nu=(\pii_{m,[\pihirac{m}{e_j}]}^a)^{-1}(D_{j,[\pihirac{m}{e_j}]}^\nu)\]
where the integer $j\in\{1,\lambdadots,g\}$ satisfies the relation
\[n_j\cdots n_g\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq m<n_{j+1}\cdots n_g\lambdaangle\nu,\gammaamma_{j+1}\rhoangle,\]
by $D_{j,m}^\nu$ we denote the corresponding set $D_m^\nu$ for the quasi-ordinary surface defined by the $j$-th approximated root, and for $q>p$, $\pii_{q,p}^a:\mathbb A^3_q\lambdaongrightarrow\mathbb A^3_p$ is the projection of the jet schemes of the affine ambient space.
\varepsilonnd{Pro}
This explains in part how the singular locus of the approximated roots play a role in our problem.
The irreducible components of the jet schemes fit in natural projective systems, to which we associate a weighted graph. The vertices of the graph correspond to irreducible components, and to every vertex we attach the corresponding codimension and embedded dimension. We will prove the following result.
\betaegin{The}
The weighted graph determines and it is determined by the topological type of the singularity.
\varepsilonnd{The}
This theorem achieves one of our goals for this type of singularities: constructing a complete invariant of the singularity from its jet schemes. Note that other invariants involving
arcs and jets, like motivic zeta functions, do not determine the topological type in the case of quasi ordinary singularities, see \cite{CoGPqo} and \cite{Nuelo}.
In another direction, using Mustata's formula (\cite{Mus}), we will determine an irreducible component of an $m$-jet scheme, or equivalently a divisor on the ambient space $\mathbb A^3$, which contributes the log canonical threshold
of the pair $X\sigmaubset\mathbb A^3$ (note that the log canonical threshold for such a pair has been computed in \cite{BGG}, by looking at the poles of the motivic zeta function). This provides us
with pairs $X\sigmaubseteq\mathbb A^3$ whose log canonical threshold is not contributed by a monomial divisorial valuation. For instance, for the quasi-ordinary surface defined by $f=(z^2-x_1x_2)^2-x_1^3x_2z$,
the log canonical thresholds satisfy this property. Note that for a pair $C\sigmaubseteq\mathbb A^2$, where $C$ is a plane curve, the log canonical threshold is always contributed by a monomial valuation. See \cite{AN} and \cite{ACLM2} for the computation of the log canonical threshold for plane curves.
Along with the same ideas of \cite{Gen}, we are working to construct an embedded resolution of singularities of $X$ from the data of the graph
constructed in this paper. We think that such a resolution puts light on the resolution of singularities obtained by González Pérez in \cite{GP4}, and give in
some sense an answer to the question of Lipman (\cite{Lipman-Eq}) on the construction of a canonical resolution of singularity of a quasi-ordinary hypersurface from the
characteristic exponents.
Moreover, understanding the surface case is an important step in the understanding of the general case.
The structure of the paper is as follows. In Section \rhoef{Sec2} we introduce jet schemes. A brief exposition on quasi-ordinary singularities is given in Section
\rhoef{secQO}, together with some useful definitions at the end of the section. Section \rhoef{Sec4} is the heart of the paper; it is devoted to the study of the irreducible components of the
jet schemes of quasi-ordinary surface singularities. In Section \rhoef{Proofs} we state and proof some results which are useful but technical, and moreover we leave the proofs of some previous results, to make Section \rhoef{Sec4} more readable.
{\betaf Acknowledgments.}
We thank Pedro González Pérez and the referees, for comments and suggestions which improved substantially the presentation of this paper.
The beginning of this work was done during a stay of HC supported by the
European Research Council under the European Community’s Seventh Framework
Programme (FP7/2007-2013) / ERC Grant Agreement nr. 246903 NMNAG. She thanks Institut Mathématique de
Jussieu for hospitality.
HM is partially supported by ANR-12-JS01-0002-01 SUSI.
\sigmaection{Jet schemes}
\lambdaabel{Sec2} | 3,670 | 114,405 | en |
train | 0.29.1 | \betaegin{The}
For any $m\in\mathbb Z_{>0}$ we have that the space of $m$-jets of $X$ with center in the singular locus has the following decomposition into irreducible components
\[\pii_m^{-1}(X_{Sing})=\cup_{\nu\in F_m}C_m^\nu.\]
\varepsilonnd{The}
Hence, for
every $m \in \mathbb Z_{>0}$, we determine the irreducible components of $X_m$ with center in the singular locus of $X$. Moreover, we give a formula for the codimension of $C_m^\nu$ in Proposition \rhoef{Prop1}.
To prove the theorem we need to understand the geometry of the sets $D_m^\nu$. This is done by comparing their geometries with
the geometries of the corresponding subsets for the approximate roots. The following proposition describes this comparison.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in\sigma\cap N_j$ such that $n_1\cdots n_g\lambdaangle\nu,\gammaamma_1\rhoangle\lambdaeq m$, we have that
\[D_m^\nu=(\pii_{m,[\pihirac{m}{e_j}]}^a)^{-1}(D_{j,[\pihirac{m}{e_j}]}^\nu)\]
where the integer $j\in\{1,\lambdadots,g\}$ satisfies the relation
\[n_j\cdots n_g\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq m<n_{j+1}\cdots n_g\lambdaangle\nu,\gammaamma_{j+1}\rhoangle,\]
by $D_{j,m}^\nu$ we denote the corresponding set $D_m^\nu$ for the quasi-ordinary surface defined by the $j$-th approximated root, and for $q>p$, $\pii_{q,p}^a:\mathbb A^3_q\lambdaongrightarrow\mathbb A^3_p$ is the projection of the jet schemes of the affine ambient space.
\varepsilonnd{Pro}
This explains in part how the singular locus of the approximated roots play a role in our problem.
The irreducible components of the jet schemes fit in natural projective systems, to which we associate a weighted graph. The vertices of the graph correspond to irreducible components, and to every vertex we attach the corresponding codimension and embedded dimension. We will prove the following result.
\betaegin{The}
The weighted graph determines and it is determined by the topological type of the singularity.
\varepsilonnd{The}
This theorem achieves one of our goals for this type of singularities: constructing a complete invariant of the singularity from its jet schemes. Note that other invariants involving
arcs and jets, like motivic zeta functions, do not determine the topological type in the case of quasi ordinary singularities, see \cite{CoGPqo} and \cite{Nuelo}.
In another direction, using Mustata's formula (\cite{Mus}), we will determine an irreducible component of an $m$-jet scheme, or equivalently a divisor on the ambient space $\mathbb A^3$, which contributes the log canonical threshold
of the pair $X\sigmaubset\mathbb A^3$ (note that the log canonical threshold for such a pair has been computed in \cite{BGG}, by looking at the poles of the motivic zeta function). This provides us
with pairs $X\sigmaubseteq\mathbb A^3$ whose log canonical threshold is not contributed by a monomial divisorial valuation. For instance, for the quasi-ordinary surface defined by $f=(z^2-x_1x_2)^2-x_1^3x_2z$,
the log canonical thresholds satisfy this property. Note that for a pair $C\sigmaubseteq\mathbb A^2$, where $C$ is a plane curve, the log canonical threshold is always contributed by a monomial valuation. See \cite{AN} and \cite{ACLM2} for the computation of the log canonical threshold for plane curves.
Along with the same ideas of \cite{Gen}, we are working to construct an embedded resolution of singularities of $X$ from the data of the graph
constructed in this paper. We think that such a resolution puts light on the resolution of singularities obtained by González Pérez in \cite{GP4}, and give in
some sense an answer to the question of Lipman (\cite{Lipman-Eq}) on the construction of a canonical resolution of singularity of a quasi-ordinary hypersurface from the
characteristic exponents.
Moreover, understanding the surface case is an important step in the understanding of the general case.
The structure of the paper is as follows. In Section \rhoef{Sec2} we introduce jet schemes. A brief exposition on quasi-ordinary singularities is given in Section
\rhoef{secQO}, together with some useful definitions at the end of the section. Section \rhoef{Sec4} is the heart of the paper; it is devoted to the study of the irreducible components of the
jet schemes of quasi-ordinary surface singularities. In Section \rhoef{Proofs} we state and proof some results which are useful but technical, and moreover we leave the proofs of some previous results, to make Section \rhoef{Sec4} more readable.
{\betaf Acknowledgments.}
We thank Pedro González Pérez and the referees, for comments and suggestions which improved substantially the presentation of this paper.
The beginning of this work was done during a stay of HC supported by the
European Research Council under the European Community’s Seventh Framework
Programme (FP7/2007-2013) / ERC Grant Agreement nr. 246903 NMNAG. She thanks Institut Mathématique de
Jussieu for hospitality.
HM is partially supported by ANR-12-JS01-0002-01 SUSI.
\sigmaection{Jet schemes}
\lambdaabel{Sec2}
In this section we define jet schemes of an affine scheme $X,$ see \cite{Ishii-07} for details.
Let $X=\mbox{Spec }\mathbb C[x_1,\lambdadots,x_n]/I$ be an affine scheme of finite type. For $m\in\mathbb Z_{>0}$ the functor
$F_m:\ \mathbb C\mbox{-Schemes}\lambdaongrightarrow \mbox{Sets}$ which, with an affine scheme defined by a $\mathbb C$-algebra $A$, associates
\[F_m(\mbox{Spec}(A))=\mbox{Hom}_\mathbb C(\mbox{Spec}(A[t]/(t^{m+1})),X),\]
is representable by a $\mathbb C$-scheme, denoted by $X_m$. This is the scheme of $m$-jets. Its closed points are morphisms of the form
\[\gammaamma:\mbox{Spec}(\mathbb C[t]/(t^{m+1}))\lambdaongrightarrow X.\]
Such a morphism $\gammaamma$ is equivalent to a $\mathbb C$-algebra homomorphism
\[\gammaamma^\alphast:\mathbb C[x_1,\lambdadots,x_n]/I\lambdaongrightarrow \mathbb C[t]/(t^{m+1}).\]
If we fix a set of generators $f_1,\lambdadots,f_r$ for the ideal $I$, the map $\gammaamma^\alphast$ is determined by the image of the $x_i^{'s}$
\[x_i\mapsto x_i^{(0)}+x_i^{(1)}t+\cdots+x_i^{(m)}t^m,\ 1\lambdaeq i\lambdaeq n,\]
where the relations
\betaegin{equation}
f_i(x_1^{(0)}+\cdots+x_1^{(m)}t^m,\lambdadots,x_n^{(0)}+\cdots+x_n^{(m)}t^m)\varepsilonquiv 0\mbox{ mod }t^{m+1}
\lambdaabel{relation}
\varepsilonnd{equation}
must hold for each $f_i$, with $1\lambdaeq i\lambdaeq r$.
If we write equations in (\rhoef{relation}) as
\betaegin{equation}
\betaegin{array}{cc}
f_i(x_1^{(0)}+x_1^{(1)}t+\cdots+x_1^{(m)}t^m,\lambdadots,x_n^{(0)}+x_n^{(1)}t+\cdots+x_n^{(m)}t^m)=\\
\\
=\sigmaum_{j=0}^mF_i^{(j)}(x_1^{(0)},\lambdadots,x_1^{(j)},\lambdadots,
x_n^{(0)},\lambdadots,x_n^{(j)})\ t^j\mbox{ mod }t^{m+1},\\
\varepsilonnd{array}
\lambdaabel{expf}
\varepsilonnd{equation}
we have that giving a closed point of $X_m$ is equivalent to giving a point in $V(F_l^{(j)})_{0\lambdaeq j\lambdaeq m,1\lambdaeq l\lambdaeq r}\sigmaubset\mathbb A_m^n$, where
$\mathbb A_m^n=\mbox{Spec}(\mathbb C[x_i^{(0)},\lambdadots,x_i^{(m)}]_{i=1,\lambdadots,n})$. Hence we can make the
following identification
\betaegin{equation}
X_m=\mbox{Spec }\lambdaeft(\pihirac{\mathbb C[x_i^{(0)},\lambdadots,x_i^{(m)}]_{i=1,\lambdadots,n}}{(F_l^{(j)})_{0\lambdaeq j\lambdaeq m,\ 1\lambdaeq l\lambdaeq r}}\rhoight).
\lambdaabel{Desc1}
\varepsilonnd{equation}
\betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The equations defining the $3$-jets are
\[\betaegin{array}{ll}
F^{(0)} & = {z^{(0)}}^3-{x_1^{(0)}}^3{x_2^{(0)}}^2\\
\\
F^{(1)} & =3{z^{(0)}}^2z^{(1)}-3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(0)}}^2-2{x_1^{(0)}}^3{x_2^{(0)}}x_2^{(1)}\\
\\
F^{(2)} & =3{z^{(0)}}^2z^{(2)}+3z^{(0)}{z^{(1)}}^2-6{x_1^{(0)}}^2{x_1^{(1)}}x_2^{(0)}x_2^{(1)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(2)}-3{x_1^{(0)}}^2x_1^{(2)}{x_2^{(0)}}^2\\
& \ \ -{x_1^{(0)}}^3{x_2^{(1)}}^2-3x_1^{(0)}{x_1^{(1)}}^2{x_2^{(0)}}^2\\
\\
F^{(3)} & = {z^{(1)}}^3+6z^{(0)}z^{(1)}z^{(2)}+3{z^{(0)}}^2z^{(3)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(3)}-2{x_1^{(0)}}^3x_2^{(1)}x_2^{(2)}-6{x_1^{(0)}}^2x_1^{(1)}x_2^{(0)}x_2^{(2)}\\
& \ \ -3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(1)}}^2-6{x_1^{(0)}}^2x_1^{(2)}x_2^{(0)}x_2^{(1)}-6x_1^{(0)}{x_1^{(1)}}^2x_2^{(0)}x_2^{(1)}-3{x_1^{(0)}}^2x_1^{(3)}{x_2^{(0)}}^2\\
& \ \ -6x_1^{(0)}x_1^{(1)}x_1^{(2)}{x_2^{(0)}}^2-{x_1^{(1)}}^3{x_2^{(0)}}^2\\
\varepsilonnd{array}\]
\lambdaabel{Ex0}
\varepsilonnd{Exam}
\betaegin{Rem}
Every polynomial $F^{(l)}$ is non-zero and quasi-homogeneous of degree $l$ in the variables $x_i^{(0)},\lambdadots,x_i^{(l)}$, $z^{(0)},\lambdadots,z^{(l)}$ for $i=1,2$. In $F^{(0)},\lambdadots,F^{(l)}$ the variables $x_1^{(l)},\ x_2^{(l)}$ and $z^{(l)}$ appear only in $F^{(l)}$.
\lambdaabel{RemTonto}
\varepsilonnd{Rem}
\sigmaection{Quasi-ordinary surface singularities}
\lambdaabel{secQO}
In this section we collect some well known facts about quasi-ordinary hypersurface singularities of dimension two, and we prove some lemmas which will be used in the next section. We state everything for the case of dimension two, though the definitions and results hold in any dimension.
An equidimensional germ $(X, 0)$, of dimension $2$, is {\varepsilonm quasi-ordinary} (q.o. for short) if there exists a finite projection $p: (X, 0) \rhoightarrow (\mathbb C^2,0)$ which is a local isomorphism outside a normal crossing divisor. If $(X, 0)$ is a hypersurface there is an embedding $(X, 0) \sigmaubset (\mathbb C^{3},0) $, where $X$ is
defined by an equation $f= 0$, and $f \in \mathbb C \{ x_1,x_2 \} [z]$ is a {\it quasi-ordinary ~polynomial}; that is, a Weierstrass polynomial with discriminant
$\Delta_z f$ of the form $\Delta_z f = x_1^{\delta_1}\cdot x_2^{\delta_2} \varepsilonpsilon$ for a unit $\varepsilonpsilon$ in the ring $ \mathbb C \{ x_1,x_2 \}$ of convergent power series and
$(\delta_1,\delta_2) \in \mathbb Z^2_{\gammaeqslant 0}$. In these coordinates the projection $p$ is the restriction of the projection
\[\mathbb C^{3}\rhoightarrow\mathbb C^2,\ \ \ \ (x_1,x_2,z)\mapsto(x_1,x_2).\]
From now on we assume that $(X,0)$ is analytically irreducible, that is $f \in \mathbb C \{ x_1, x_2 \} [z]$ is irreducible (see \cite{Ass} and \cite{Ev} for criteria of irreducibility of q.o. polynomial). The Jung-Abhyankar theorem guarantees
that the roots of a q.o.~ polynomial $f$, called {\it q.o.~branches}, are fractional power series in $\mathbb C \{ x_1^{1/n},x_2^{1/n}\}$, for $n =\deltaeg f$ (see
\cite{Abhyankar}). The difference $\zetaeta^{(i)}-\zetaeta^{(j)}$ of two different roots of $f$ divides the discriminant of $f$ in the ring $\mathbb C\{x_1^{1/n},
x_2^{1/n}\}$. Therefore $\zetaeta^{(i)}-\zetaeta^{(j)}=x_1^{\lambda_{ij}^{(1)}}x_2^{\lambda_{ij}^{(2)}}u_{ij}$ where $u_{ij}$ is a unit in $\mathbb C\{x_1^{1/n},x_2^{1/n}\}$. The exponents $\lambda_{ij}$
are characterized in the following Lemma: | 3,858 | 114,405 | en |
train | 0.29.2 | \betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The equations defining the $3$-jets are
\[\betaegin{array}{ll}
F^{(0)} & = {z^{(0)}}^3-{x_1^{(0)}}^3{x_2^{(0)}}^2\\
\\
F^{(1)} & =3{z^{(0)}}^2z^{(1)}-3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(0)}}^2-2{x_1^{(0)}}^3{x_2^{(0)}}x_2^{(1)}\\
\\
F^{(2)} & =3{z^{(0)}}^2z^{(2)}+3z^{(0)}{z^{(1)}}^2-6{x_1^{(0)}}^2{x_1^{(1)}}x_2^{(0)}x_2^{(1)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(2)}-3{x_1^{(0)}}^2x_1^{(2)}{x_2^{(0)}}^2\\
& \ \ -{x_1^{(0)}}^3{x_2^{(1)}}^2-3x_1^{(0)}{x_1^{(1)}}^2{x_2^{(0)}}^2\\
\\
F^{(3)} & = {z^{(1)}}^3+6z^{(0)}z^{(1)}z^{(2)}+3{z^{(0)}}^2z^{(3)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(3)}-2{x_1^{(0)}}^3x_2^{(1)}x_2^{(2)}-6{x_1^{(0)}}^2x_1^{(1)}x_2^{(0)}x_2^{(2)}\\
& \ \ -3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(1)}}^2-6{x_1^{(0)}}^2x_1^{(2)}x_2^{(0)}x_2^{(1)}-6x_1^{(0)}{x_1^{(1)}}^2x_2^{(0)}x_2^{(1)}-3{x_1^{(0)}}^2x_1^{(3)}{x_2^{(0)}}^2\\
& \ \ -6x_1^{(0)}x_1^{(1)}x_1^{(2)}{x_2^{(0)}}^2-{x_1^{(1)}}^3{x_2^{(0)}}^2\\
\varepsilonnd{array}\]
\lambdaabel{Ex0}
\varepsilonnd{Exam}
\betaegin{Rem}
Every polynomial $F^{(l)}$ is non-zero and quasi-homogeneous of degree $l$ in the variables $x_i^{(0)},\lambdadots,x_i^{(l)}$, $z^{(0)},\lambdadots,z^{(l)}$ for $i=1,2$. In $F^{(0)},\lambdadots,F^{(l)}$ the variables $x_1^{(l)},\ x_2^{(l)}$ and $z^{(l)}$ appear only in $F^{(l)}$.
\lambdaabel{RemTonto}
\varepsilonnd{Rem}
\sigmaection{Quasi-ordinary surface singularities}
\lambdaabel{secQO}
In this section we collect some well known facts about quasi-ordinary hypersurface singularities of dimension two, and we prove some lemmas which will be used in the next section. We state everything for the case of dimension two, though the definitions and results hold in any dimension.
An equidimensional germ $(X, 0)$, of dimension $2$, is {\varepsilonm quasi-ordinary} (q.o. for short) if there exists a finite projection $p: (X, 0) \rhoightarrow (\mathbb C^2,0)$ which is a local isomorphism outside a normal crossing divisor. If $(X, 0)$ is a hypersurface there is an embedding $(X, 0) \sigmaubset (\mathbb C^{3},0) $, where $X$ is
defined by an equation $f= 0$, and $f \in \mathbb C \{ x_1,x_2 \} [z]$ is a {\it quasi-ordinary ~polynomial}; that is, a Weierstrass polynomial with discriminant
$\Delta_z f$ of the form $\Delta_z f = x_1^{\delta_1}\cdot x_2^{\delta_2} \varepsilonpsilon$ for a unit $\varepsilonpsilon$ in the ring $ \mathbb C \{ x_1,x_2 \}$ of convergent power series and
$(\delta_1,\delta_2) \in \mathbb Z^2_{\gammaeqslant 0}$. In these coordinates the projection $p$ is the restriction of the projection
\[\mathbb C^{3}\rhoightarrow\mathbb C^2,\ \ \ \ (x_1,x_2,z)\mapsto(x_1,x_2).\]
From now on we assume that $(X,0)$ is analytically irreducible, that is $f \in \mathbb C \{ x_1, x_2 \} [z]$ is irreducible (see \cite{Ass} and \cite{Ev} for criteria of irreducibility of q.o. polynomial). The Jung-Abhyankar theorem guarantees
that the roots of a q.o.~ polynomial $f$, called {\it q.o.~branches}, are fractional power series in $\mathbb C \{ x_1^{1/n},x_2^{1/n}\}$, for $n =\deltaeg f$ (see
\cite{Abhyankar}). The difference $\zetaeta^{(i)}-\zetaeta^{(j)}$ of two different roots of $f$ divides the discriminant of $f$ in the ring $\mathbb C\{x_1^{1/n},
x_2^{1/n}\}$. Therefore $\zetaeta^{(i)}-\zetaeta^{(j)}=x_1^{\lambda_{ij}^{(1)}}x_2^{\lambda_{ij}^{(2)}}u_{ij}$ where $u_{ij}$ is a unit in $\mathbb C\{x_1^{1/n},x_2^{1/n}\}$. The exponents $\lambda_{ij}$
are characterized in the following Lemma:
\betaegin{Lem} \lambdaabel{expo} {\rhom (see \cite{Gau}, Prop. 1.3)}
Let $f \in \mathbb C \{ x_1, x_2 \} [z] $ be an irreducible
q.o.~polynomial. Let $\zeta$ be a root of $f$ with expansion:
\betaegin{equation} \lambdaabel{expan}
\zeta = \sigmaum \betaeta_{\lambda} \betaold{x}^\lambda.
\varepsilonnd{equation}
There exists $0 \ne \lambda_1, \deltaots, \lambda_g \in \mathbb Q^2_{\gammaeqslant 0}$
such that $\lambda_1\lambdaeq\lambda_2\lambdaeq\cdots\lambdaeq\lambda_g$, and if $M_0 :=\mathbb Z^2 $ and $M_j := M_{j-1} + \mathbb Z \lambda_j$ for
$j=1, \deltaots, g$, then:
\betaegin{enumerate}
\item [(i)] $\betaeta_{\lambda_i} \ne 0$ and if $\betaeta_{\lambda} \ne 0$ then $\lambda \in M_j$ where
$j$ is the unique integer such that $\lambda_j \lambdaeqslant \lambda$ and
$\lambda_{j+1} \nleq \lambda$ (where $\lambdaeqslant$ means coordinate-wise and we convey that $\lambda_{g +1} = \infty$).
\item [(ii)] For $j=1, \deltaots, g$, we have $\lambda_j \notin M_{j-1}$, hence the index
$n_j = [M_{j-1} : M_j]$ is $> 1$.
\varepsilonnd{enumerate}
Moreover if $\zetaeta\in\mathbb C\{x_1^{1/n},x_2^{1/n}\}$ is a fractional power series satisfying the conditions above, then $\zetaeta$ is a quasi-ordinary branch.
\varepsilonnd{Lem}
\betaegin{Defi}
The exponents $\lambda_1 , \deltaots, \lambda_g $ in Lemma \rhoef{expo} are called {\varepsilonm characteristic exponents} of the q.o.~branch $\zeta$. We denote by $M$ the lattice
$M_g$ and we call it the lattice associated to the q.o.~branch $\zeta$. We denote by $N$ (resp. $N_i$) the dual lattice of $M$ (resp. $M_i$ for $i=1,\lambdadots,g$). For
convenience we denote $\lambda_0:=(0,0)$ and $n_0:=1$.
\lambdaabel{defExp}
\varepsilonnd{Defi}
In \cite{Gau} Gau proved that the characteristic exponents determine and are determined by the embedded topological type of $(X,0)$.
As a consequence of Lemma \rhoef{expo} we have the following result:
\betaegin{Lem}
If $\zetaeta$ is a quasi-ordinary branch of the form (\rhoef{expan}) then the series $\zetaeta_{j-1}:=\sigmaum_{\lambdaambda\not\gammaeq\lambdaambda_j}\betaeta_\lambdaambda\betaold{x}^\lambdaambda$ is a
quasi-ordinary branch with characteristic exponents $\lambda_1,\lambdadots,\lambda_{j-1}$, for $j=1,\lambdadots,g$.
\lambdaabel{LemConseq}
\varepsilonnd{Lem}
\betaegin{Defi}
For $0\lambdaeq j\lambdaeq g-1$ we have the germ of quasi-ordinary hypersurface $(X^{(j)},0)$, where $X^{(j)}$ is parametrized by the branch $\zetaeta_j$. For convenience we also denote $\zetaeta$ by $\zetaeta_g$ and $X$ by $X^{(g)}$.
\lambdaabel{SemiX}
\varepsilonnd{Defi}
Without loss of generality we relabel the variables $x_1, x_2$ in such a way that if $\lambda_j = ( \lambda_j^{(1)}, \lambda_j^{(2)}) \in
\mathbb Q^2$ for $j=1, \deltaots, g$, then we have:
\betaegin{equation}
(\lambda_1^{(1)}, \deltaots, \lambda_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}} (\lambda_1^{(2)}, \deltaots, \lambda_g^{(2)}),
\lambdaabel{lex}
\varepsilonnd{equation}
where $\gammaeqslant_{\mbox{\rhom lex}}$ is lexicographic order. The
q.o.~branch $\zeta$
is said to be normalized if
$\lambda_1$ is not of the form $(\lambda_1^{(1)}, 0)$ with $\lambda_1^{(1)} <
1$. Lipman proved that the germ $(X,0)$ can be parametrized by a
normalized q.o.~branch (see \cite{Gau}, Appendix).
We assume from now on that the q.o.~branch $\zeta$ is normalized.
The semigroup $\mathbb Z^2_{\gammaeqslant 0}$ has a minimal set of generators
$v_1,v_2$, which is a basis of the lattice $M_0$. The dual
basis, $\{w_1,w_2\}$, is a basis of the dual lattice $N_0$, and spans a regular cone $\sigma$ in
$N_{0,\mathbb R}=N_0\mathbb otimes_\mathbb Z\mathbb R$. It follows that $\mathbb Z^2_{\gammaeqslant 0} = \sigma^\vee \cap
M_0$, where $\sigma^\vee = \mathbb R^2_{\gammaeqslant 0}$ is the dual cone of
$\sigma$. The $\mathbb C$-algebra $\mathbb C \{ x_1,x_2 \} $ is isomorphic
to $\mathbb C \{ \sigma^\vee \cap M_0 \} $. The
local algebra ${\mathcal O}_X = \mathbb C\{x_1, x_2\}[z]/(f)$ of the singularity $(X,0)$ is isomorphic to
$\mathbb C\{\sigma^\vee \cap M_0\}[\zeta]$. By Lemma \rhoef{expo} the series $\zeta
$ can be viewed as an element $\sigmaum \betaeta_\lambda \betaold x^\lambda$ of the algebra
$ \mathbb C\{\sigma^\vee \cap M\}$.
\betaegin{Lem} (see \cite{GP4})
The homomorphism $\mathcal O_X\lambdaongrightarrow\mathbb C\{\sigma^\vee\cap M\}$ is the inclusion of $\mathcal O_X$ in its integral closure in its field of fractions.
\lambdaabel{NormTor}
\varepsilonnd{Lem}
This Lemma shows that the normalization of a quasi-ordinary hypersurface $(X,0)$ is the germ of the toric variety $X(\sigma,N)=Z^{\sigma^\vee\cap M}$ at the distinguished point.
The elements of $M$ defined by:
\betaegin{equation}
{\gamma}_1 = \lambda_1\mbox{ and } {\gamma}_{j+1}- n_j {\gamma}_{j} = \lambda_{j+1} - \lambda_{j}\mbox{ for } j= 1, \deltaots, g-1,
\lambdaabel{defSem}
\varepsilonnd{equation}
span the semigroup $\Gamma := \mathbb Z^2_{\gammaeqslant 0} + \gamma_1 \mathbb Z_{\gammaeqslant
0} + \cdots + \gamma_g \mathbb Z_{\gammaeqslant 0} \sigmaubset \sigma^\vee \cap M$. For convenience we denote $\gammaamma_0:=0$. The
semigroup $\Gamma$ defines an analytic invariant of the germ
$(X,0)$ (see \cite{Jussieu},\cite{PPP04},\cite{KM}).
\betaegin{Defi}
The monomial variety associated to $(X,0)$ is the toric variety
\[X^\Gamma:=\mbox{Spec }\mathbb C[\Gamma].\]
\lambdaabel{defMV}
\varepsilonnd{Defi}
Moreover we associate with the characteristic exponents the following sequence of semigroups:
\[\Gamma_j=\sigma^\vee\cap M+\gammaamma_1\mathbb Z_{\gammaeq 0}+\cdots+\gammaamma_j\mathbb Z_{\gammaeq 0},\mbox{ for }j=0,\lambdadots,g.\]
And we have the corresponding monomial varieties associated to $\Gamma_j$. We denote by $e_{i-1}:=n_i\cdots n_g$ for $1<i\lambdaeq g$ and set $e_g:=1$.
Notice that, by (\rhoef{lex}) and the definition of $\gammaamma_1,\lambdadots,\gammaamma_g$, we deduce that
\betaegin{equation}
(\gammaamma_1^{(1)}, \deltaots, \gammaamma_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}}(\gammaamma_1^{(2)}, \deltaots, \gammaamma_g^{(2)}).
\lambdaabel{lexGamma}
\varepsilonnd{equation}
The following Lemma gathers some important facts about the generators $\gammaamma_j$ and the semigroups $\Gamma_j$. | 3,739 | 114,405 | en |
train | 0.29.3 | \lambdaabel{LemConseq}
\varepsilonnd{Lem}
\betaegin{Defi}
For $0\lambdaeq j\lambdaeq g-1$ we have the germ of quasi-ordinary hypersurface $(X^{(j)},0)$, where $X^{(j)}$ is parametrized by the branch $\zetaeta_j$. For convenience we also denote $\zetaeta$ by $\zetaeta_g$ and $X$ by $X^{(g)}$.
\lambdaabel{SemiX}
\varepsilonnd{Defi}
Without loss of generality we relabel the variables $x_1, x_2$ in such a way that if $\lambda_j = ( \lambda_j^{(1)}, \lambda_j^{(2)}) \in
\mathbb Q^2$ for $j=1, \deltaots, g$, then we have:
\betaegin{equation}
(\lambda_1^{(1)}, \deltaots, \lambda_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}} (\lambda_1^{(2)}, \deltaots, \lambda_g^{(2)}),
\lambdaabel{lex}
\varepsilonnd{equation}
where $\gammaeqslant_{\mbox{\rhom lex}}$ is lexicographic order. The
q.o.~branch $\zeta$
is said to be normalized if
$\lambda_1$ is not of the form $(\lambda_1^{(1)}, 0)$ with $\lambda_1^{(1)} <
1$. Lipman proved that the germ $(X,0)$ can be parametrized by a
normalized q.o.~branch (see \cite{Gau}, Appendix).
We assume from now on that the q.o.~branch $\zeta$ is normalized.
The semigroup $\mathbb Z^2_{\gammaeqslant 0}$ has a minimal set of generators
$v_1,v_2$, which is a basis of the lattice $M_0$. The dual
basis, $\{w_1,w_2\}$, is a basis of the dual lattice $N_0$, and spans a regular cone $\sigma$ in
$N_{0,\mathbb R}=N_0\mathbb otimes_\mathbb Z\mathbb R$. It follows that $\mathbb Z^2_{\gammaeqslant 0} = \sigma^\vee \cap
M_0$, where $\sigma^\vee = \mathbb R^2_{\gammaeqslant 0}$ is the dual cone of
$\sigma$. The $\mathbb C$-algebra $\mathbb C \{ x_1,x_2 \} $ is isomorphic
to $\mathbb C \{ \sigma^\vee \cap M_0 \} $. The
local algebra ${\mathcal O}_X = \mathbb C\{x_1, x_2\}[z]/(f)$ of the singularity $(X,0)$ is isomorphic to
$\mathbb C\{\sigma^\vee \cap M_0\}[\zeta]$. By Lemma \rhoef{expo} the series $\zeta
$ can be viewed as an element $\sigmaum \betaeta_\lambda \betaold x^\lambda$ of the algebra
$ \mathbb C\{\sigma^\vee \cap M\}$.
\betaegin{Lem} (see \cite{GP4})
The homomorphism $\mathcal O_X\lambdaongrightarrow\mathbb C\{\sigma^\vee\cap M\}$ is the inclusion of $\mathcal O_X$ in its integral closure in its field of fractions.
\lambdaabel{NormTor}
\varepsilonnd{Lem}
This Lemma shows that the normalization of a quasi-ordinary hypersurface $(X,0)$ is the germ of the toric variety $X(\sigma,N)=Z^{\sigma^\vee\cap M}$ at the distinguished point.
The elements of $M$ defined by:
\betaegin{equation}
{\gamma}_1 = \lambda_1\mbox{ and } {\gamma}_{j+1}- n_j {\gamma}_{j} = \lambda_{j+1} - \lambda_{j}\mbox{ for } j= 1, \deltaots, g-1,
\lambdaabel{defSem}
\varepsilonnd{equation}
span the semigroup $\Gamma := \mathbb Z^2_{\gammaeqslant 0} + \gamma_1 \mathbb Z_{\gammaeqslant
0} + \cdots + \gamma_g \mathbb Z_{\gammaeqslant 0} \sigmaubset \sigma^\vee \cap M$. For convenience we denote $\gammaamma_0:=0$. The
semigroup $\Gamma$ defines an analytic invariant of the germ
$(X,0)$ (see \cite{Jussieu},\cite{PPP04},\cite{KM}).
\betaegin{Defi}
The monomial variety associated to $(X,0)$ is the toric variety
\[X^\Gamma:=\mbox{Spec }\mathbb C[\Gamma].\]
\lambdaabel{defMV}
\varepsilonnd{Defi}
Moreover we associate with the characteristic exponents the following sequence of semigroups:
\[\Gamma_j=\sigma^\vee\cap M+\gammaamma_1\mathbb Z_{\gammaeq 0}+\cdots+\gammaamma_j\mathbb Z_{\gammaeq 0},\mbox{ for }j=0,\lambdadots,g.\]
And we have the corresponding monomial varieties associated to $\Gamma_j$. We denote by $e_{i-1}:=n_i\cdots n_g$ for $1<i\lambdaeq g$ and set $e_g:=1$.
Notice that, by (\rhoef{lex}) and the definition of $\gammaamma_1,\lambdadots,\gammaamma_g$, we deduce that
\betaegin{equation}
(\gammaamma_1^{(1)}, \deltaots, \gammaamma_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}}(\gammaamma_1^{(2)}, \deltaots, \gammaamma_g^{(2)}).
\lambdaabel{lexGamma}
\varepsilonnd{equation}
The following Lemma gathers some important facts about the generators $\gammaamma_j$ and the semigroups $\Gamma_j$.
\betaegin{Lem}(see Lemma 3.3 in \cite{Jussieu})
\betaegin{enumerate}
\item[(i)] We have that $\gammaamma_j>n_{j-1}\gammaamma_{j-1}$ for $j=2,\lambdadots,g$, where $<$ means $\neq$ and $\lambdaeq$ coordinate-wise.
\\
\item[(ii)] If a vector $u_j\in\sigma^\vee\cap M_j$, then we have $u_j+n_j\gammaamma_j\in\Gamma_j$.
\\
\item[(iii)] The vector $n_j\gammaamma_j$ belongs to the semigroup $\Gamma_{j-1}$ for $j=1,\lambdadots,g$. Moreover, we have a unique relation
\betaegin{equation}
n_j\gammaamma_j=\alphalpha^{(j)}+r_1^{(j)}\gammaamma_1+\cdots+r_{j-1}^{(j)}\gammaamma_{j-1}
\lambdaabel{relgam}
\varepsilonnd{equation}
such that $0\lambdaeq r_i^{(j)}\lambdaeq n_i-1$ and $\alphalpha^{(j)}\in M_0$ for $j=1,\lambdadots,g$.
\varepsilonnd{enumerate}
\lambdaabel{LemaPedro}
\varepsilonnd{Lem}
\betaegin{Defi}
Given two irreducible quasi-ordinary polynomials $f$ and $g$ in $\mathbb C\{x_1,x_2\}[z]$ such that $fg$ is a quasi-ordinary polynomial, we say that $f$ and $g$
have order of coincidence $\alphalpha\in \mathbb Q^2$ if $\alphalpha$ is the largest exponent on the set
\[\{\lambda_{ij}\ |\ f(\zetaeta^{(i)})=g(\zetaeta^{(j)})=0\},\]
where $\zetaeta^{(i)}$ and $\zetaeta^{(j)}$ are roots of $fg$.
\lambdaabel{OrdCon}
\varepsilonnd{Defi}
\betaegin{Defi}
We associate to $f$ a set of semi-roots
\[z=f_0,f_1\lambdadots,f_g=f\in \mathbb C\{x_1,x_2\}[z].\]
Every $f_j$ is an irreducible quasi-ordinary polynomial of degree $n_0\cdots n_j$ with order of coincidence with $f$ equal to $\lambda_{j+1}$ for $j=0,\lambdadots,g$.
\lambdaabel{defAroots}
\varepsilonnd{Defi}
They are parametrized by truncations of a root $\zetaeta(x_1^{1/n},x_2^{1/n})$ of $f$ in the following sense:
\betaegin{Pro}
(see \cite{Jussieu}) Let $q\in \mathbb C\{x_1,x_2\}[z]$ be a monic polynomial of degree $n_0\cdots n_j$. Then $q$ is a $j$-th semi-root of $f$ if and only if
$q(\zetaeta)=\betaold{x}^{\gammaamma_j}\varepsilonpsilon_j$ for a unit $\varepsilonpsilon_j$ in $\mathbb C\{x_1,x_2\}[z]$.
\varepsilonnd{Pro}
\betaegin{Cor}
The quasi-ordinary polynomials $f_j\in\mathbb C\{x_1,x_2\}[z]$ defining $X^{(j)}$ (see Definition \rhoef{SemiX}) for $j=0,\lambdadots,g$ form a system of semiroots of $f$.
\lambdaabel{CorSR}
\varepsilonnd{Cor}
In what follows we state some results about quasi-ordinary polynomials and approximated roots. Moreover we give some definitions and notations that will be
used in the next section.
Approximated roots play an important role in the understanding of quasi-ordinary singularities. We have the following expansions of the semiroots in terms of the previous ones:
\betaegin{Lem} (See Lemma 35 in \cite{GP4})
The expansion of the approximated roots is of the following form:
\betaegin{equation}
c_j^*f_j=f_{j-1}^{n_j}-c_jx_1^{\alphalpha_1^{(j)}}x_2^{\alphalpha_2^{(j)}}f_0^{r_1^{(j)}}\cdots f_{j-2}^{r_{j-1}^{(j)}}+\sigmaum c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}f_0^{r_1}\cdots f_{j-1}^{r_j},
\lambdaabel{expSR}
\varepsilonnd{equation}
where $c_j^*,c_j\in\mathbb C^*$, $0\lambdaeq r_i^{(j)},r_i<n_i$ for $i=1,\lambdadots,j$, and \[n_j\gammaamma_j=(\alphalpha_1^{(j)},\alphalpha_2^{(j)})+r_1^{(j)}\gammaamma_1+\cdots+r_{j-1}^{(j)}\gammaamma_{j-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j.\]
\lambdaabel{Lema35}
\varepsilonnd{Lem}
Let $(X,0)\sigmaubset(\mathbb C^3,0)$ be a germ of quasi-ordinary surface with characteristic exponents $\lambda_1,\lambdadots,\lambda_g$. We denote by $\lambda_1=(\pihirac{a_1}{n_1},\pihirac{b_1}{n_1})$ the first
characteristic exponent.
Notice that, by (\rhoef{lex}), we have that $a_1\gammaeq b_1\gammaeq 0$.
\betaegin{Lem}
We have that
\betaegin{equation}
f_1=z^{n_1}-c_1x_1^{a_1}x_2^{b_1}+\sigmaum c_{ijk}x_1^ix_2^jz^k,
\lambdaabel{eqSR1}
\varepsilonnd{equation}
with $(i,j)+k\gammaamma_1>n_1\gammaamma_1$ and $k<n_1$ whenever $c_{ijk}\neq 0$.
And for $1\lambdaeq l\lambdaeq g-1$ we have
\betaegin{equation}
f=f_l^{e_l}+\sigmaum_{(i,j)+k\gammaamma_1>n_le_l\gammaamma_l}c_{ijk}^{(l)}x_1^ix_2^jz^k.
\lambdaabel{eqSRpqprima}
\varepsilonnd{equation}
Moreover, the following expansions will be useful. For $0\lambdaeq j<g-1$
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}
\lambdaabel{grgr1}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}>n_{j+1}e_{j+1}\gammaamma_{j+1}$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j-1}^{r_j}
\lambdaabel{grgr2}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j>n_je_j\gammaamma_j$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\lambdaabel{LemExpSR}
\varepsilonnd{Lem}
{\varepsilonm Proof.} By applying recursively Lemma \rhoef{Lema35} and using Lemma \rhoef{LemaPedro} (i).\pisifill $\Box$
\sigmaubsection{Some toric geometry}
\lambdaabel{toric}
See \cite{F} for a reference on toric geometry.
Given a lattice $N$ we denote by $N_\mathbb R$ the vector space spanned by $N$ over the field $\mathbb R$. We denote by $M$ the dual lattice, $M=\mbox{Hom}(N,\mathbb Z)$, and by $\lambdaangle,\rhoangle : N\tauimes M\lambdaongrightarrow\mathbb Z$ the duality pairing between the lattices $N$ and $M$.
A rational convex polyhedral cone (or simply a cone) is the set of non-negative linear combinations of vectors $v_1,\lambdadots,v_r\in N$. A cone is strictly convex if it contains no lines. The dual cone of $\sigma$, denoted by $\sigma^\vee$, is the set
\[\sigma^\vee=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle\gammaeq 0\ \pihiorall u\in\sigma\},\]
and the orthogonal of $\sigma$, denoted by $\sigma^\betaot$, is
\[\sigma^\betaot=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle=0\ \pihiorall u\in\sigma\}.\]
We denote by $\sigmatackrel{\circ}{\sigma}$ the relative interior of the cone $\sigma$. A fan $\Sigma$ is a family of strictly convex cones in $N_\mathbb R$ such that for any $\sigma\in\Sigma$ any face of $\sigma$ belongs to $\Sigma$, and for any $\sigma,\tau\in\Sigma$, the intersection $\sigma\cap\tau$ is a face of both.
The relation $\tau\lambdaeq\sigma$ denotes that $\tau$ is a face of $\sigma$. The support of the fan $\Sigma$ is the set $|\Sigma|:=\cup_{\tau\in\Sigma}\tau\sigmaubset N_\mathbb R$. | 3,936 | 114,405 | en |
train | 0.29.4 | \betaegin{Lem} (See Lemma 35 in \cite{GP4})
The expansion of the approximated roots is of the following form:
\betaegin{equation}
c_j^*f_j=f_{j-1}^{n_j}-c_jx_1^{\alphalpha_1^{(j)}}x_2^{\alphalpha_2^{(j)}}f_0^{r_1^{(j)}}\cdots f_{j-2}^{r_{j-1}^{(j)}}+\sigmaum c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}f_0^{r_1}\cdots f_{j-1}^{r_j},
\lambdaabel{expSR}
\varepsilonnd{equation}
where $c_j^*,c_j\in\mathbb C^*$, $0\lambdaeq r_i^{(j)},r_i<n_i$ for $i=1,\lambdadots,j$, and \[n_j\gammaamma_j=(\alphalpha_1^{(j)},\alphalpha_2^{(j)})+r_1^{(j)}\gammaamma_1+\cdots+r_{j-1}^{(j)}\gammaamma_{j-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j.\]
\lambdaabel{Lema35}
\varepsilonnd{Lem}
Let $(X,0)\sigmaubset(\mathbb C^3,0)$ be a germ of quasi-ordinary surface with characteristic exponents $\lambda_1,\lambdadots,\lambda_g$. We denote by $\lambda_1=(\pihirac{a_1}{n_1},\pihirac{b_1}{n_1})$ the first
characteristic exponent.
Notice that, by (\rhoef{lex}), we have that $a_1\gammaeq b_1\gammaeq 0$.
\betaegin{Lem}
We have that
\betaegin{equation}
f_1=z^{n_1}-c_1x_1^{a_1}x_2^{b_1}+\sigmaum c_{ijk}x_1^ix_2^jz^k,
\lambdaabel{eqSR1}
\varepsilonnd{equation}
with $(i,j)+k\gammaamma_1>n_1\gammaamma_1$ and $k<n_1$ whenever $c_{ijk}\neq 0$.
And for $1\lambdaeq l\lambdaeq g-1$ we have
\betaegin{equation}
f=f_l^{e_l}+\sigmaum_{(i,j)+k\gammaamma_1>n_le_l\gammaamma_l}c_{ijk}^{(l)}x_1^ix_2^jz^k.
\lambdaabel{eqSRpqprima}
\varepsilonnd{equation}
Moreover, the following expansions will be useful. For $0\lambdaeq j<g-1$
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}
\lambdaabel{grgr1}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}>n_{j+1}e_{j+1}\gammaamma_{j+1}$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j-1}^{r_j}
\lambdaabel{grgr2}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j>n_je_j\gammaamma_j$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\lambdaabel{LemExpSR}
\varepsilonnd{Lem}
{\varepsilonm Proof.} By applying recursively Lemma \rhoef{Lema35} and using Lemma \rhoef{LemaPedro} (i).\pisifill $\Box$
\sigmaubsection{Some toric geometry}
\lambdaabel{toric}
See \cite{F} for a reference on toric geometry.
Given a lattice $N$ we denote by $N_\mathbb R$ the vector space spanned by $N$ over the field $\mathbb R$. We denote by $M$ the dual lattice, $M=\mbox{Hom}(N,\mathbb Z)$, and by $\lambdaangle,\rhoangle : N\tauimes M\lambdaongrightarrow\mathbb Z$ the duality pairing between the lattices $N$ and $M$.
A rational convex polyhedral cone (or simply a cone) is the set of non-negative linear combinations of vectors $v_1,\lambdadots,v_r\in N$. A cone is strictly convex if it contains no lines. The dual cone of $\sigma$, denoted by $\sigma^\vee$, is the set
\[\sigma^\vee=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle\gammaeq 0\ \pihiorall u\in\sigma\},\]
and the orthogonal of $\sigma$, denoted by $\sigma^\betaot$, is
\[\sigma^\betaot=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle=0\ \pihiorall u\in\sigma\}.\]
We denote by $\sigmatackrel{\circ}{\sigma}$ the relative interior of the cone $\sigma$. A fan $\Sigma$ is a family of strictly convex cones in $N_\mathbb R$ such that for any $\sigma\in\Sigma$ any face of $\sigma$ belongs to $\Sigma$, and for any $\sigma,\tau\in\Sigma$, the intersection $\sigma\cap\tau$ is a face of both.
The relation $\tau\lambdaeq\sigma$ denotes that $\tau$ is a face of $\sigma$. The support of the fan $\Sigma$ is the set $|\Sigma|:=\cup_{\tau\in\Sigma}\tau\sigmaubset N_\mathbb R$.
Let $\tau$ be a strictly convex cone, rational for the lattice $N$. By Gordan's Lemma the semigroup $\sigma^\vee\cap M$ is finitely generated. We denote by $\mathbb C[\sigma^\vee\cap M]$ the semigroup algebra of $\sigma^\vee\cap M$ with complex coefficients. The toric variety $Z(\tau,N):=\mbox{Spec }\mathbb C[\tau^\vee\cap M]$ is normal. The torus $T_N:=Z(N)$ is an open dense subset of $Z(\tau,N)$ which acts on $Z(\tau,N)$ and the action extends the action of the torus on itself by multiplication. There is a one to one correspondence between the faces $\tauheta$ of $\tau$ and the orbits orb$(\tauheta)$ of the torus action on $Z(\tau,N)$, which reverses the inclusions of their closures. The closure of orb$(\tauheta)$ is the toric variety $Z((\sigma^\vee\cap\tauheta^\betaot)^\vee,N)$ for $\tau\lambdaeq\sigma$.
\sigmaubsection{Definitions and Notations}
\lambdaabel{defsnots}
We introduce now some definitions and notations which will be used throughout the paper.
The singular locus of a quasi-ordinary singularity is determined, after Lipman, by its characteristic exponents (see \cite{Lipman2} and \cite{PPP04}).
\betaegin{Defi} We define
\[\betaegin{array}{ll}
Z_i=X\cap\{x_i=0\}, & \mbox{for }i=1,2\\
\\
Z_{12}=X\cap\{x_1=x_2=0\}. & \\
\varepsilonnd{array}\]
Moreover, the smallest number $c\in\{1,2\}$ with the property that
\[\lambda_i^{(j)}=0,\mbox{ for all }1\lambdaeq i\lambdaeq g\mbox{ and }c+1\lambdaeq j\lambdaeq 2\]
is called the equisingular dimension of the quasi-ordinary projection $p$.
\lambdaabel{equi}
\varepsilonnd{Defi}
In \cite{Lipman2}
Lipman proved that the spaces $Z_1,Z_2$ and $Z_{12}$ are irreducible.
By condition (\rhoef{lex}) we have that $c$ gives the number of variables appearing in the monomials $\betaold x^{\lambda_1},\lambdadots,\betaold x^{\lambda_g}$.
\betaegin{Defi}
Let $X$ be a quasi-ordinary surface singularity with $g\gammaeq 1$ characteristic exponents. We define the integers $g_1\gammaeq 0$ and $g_2\in\{g_1,g_1+1\}$ as follows
\[\mbox{ if }c=1\mbox{ we set }g_1=g_2=g+1,\]
otherwise
\[\betaegin{array}{l}
\gammaamma_{g_1}^{(2)}=0\mbox{ and }\gammaamma_{g_1+1}^{(2)}\neq 0,\\
\\
g_2=\lambdaeft\{\betaegin{array}{cl}
g_1+1 & \mbox{ if }\gammaamma_{g_1+1}^{(2)}=\pihirac{1}{n_{g_1+1}}\\
\\
g_1 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\\
\varepsilonnd{array}\]
\lambdaabel{defg1}
\varepsilonnd{Defi}
Note that these integers can be defined with the same property for the characteristic exponents. Now we use them to describe the singular locus of $X$.
Lipman's theorem describes the singular locus $X_{Sing}$ of a quasi-ordinary hypersurface $X$. We state it here in the particular case of surfaces.
\betaegin{The} (See Theorem 7.3 in \cite{Lipman2})
Let $X$ be a quasi-ordinary surface singularity with characteristic exponents $\lambda_1,\lambdadots,\lambda_g$. Then we have:
\betaegin{enumerate}
\item[(i)] $X_{Sing}=Z_{12}$ if and only if $g=1$ and $\lambda_1=(\pihirac{1}{n},\pihirac{1}{n})$.
\item[(ii)] If $c=1$ then $X_{Sing}=Z_1$.
\item[(iii)] Otherwise $c=2$, and since $\lambda_1^{(1)}\neq 0$, $Z_1\sigmaubset X$ is a component of $X_{Sing}$. Moreover, if we do not have
simultaneously $\lambda_k^{(2)}=0$ for all $1\lambdaeq k\lambdaeq g-1$ and $\lambda_g^{(2)}=\pihirac{1}{n_g}$ then the singular locus is reducible of the form $X_{Sing}=Z_1\cup Z_2$.
\varepsilonnd{enumerate}
\lambdaabel{CorSing}
\varepsilonnd{The}
\betaegin{Rem}
Notice that
\[\betaegin{array}{l}
Z_1=\{x_1=z=0\}\\
\\
Z_2=\{x_2=f_{g_1}=0\}\\
\\
Z_{12}=\{(0,0,0)\}\\
\varepsilonnd{array}\]
and hence the singular locus of a quasi-ordinary surface singularity is either a point, or a line, or two lines, or a line and a singular curve.
\varepsilonnd{Rem}
Then, geometrically, the meaning of the integer $g_2$ is to measure the irreducibility of the singular locus of the approximated roots. Indeed,
\[\betaegin{array}{l}
X_{Sing}^{(j)} \mbox{ is irreducible, for }1\lambdaeq j\lambdaeq g_2\\
\\
X_{Sing}^{(j)} \mbox{ is reducible, for }g_2<j\lambdaeq g\\
\varepsilonnd{array}\]
Now we define a sequence of semi-open cones keeping track of the singular locus of the quasi-ordinary hypersurfaces $X^{(j)}$ for $j=1,\lambdadots,g$ (see Definition
\rhoef{SemiX}). Let
\[\nu_j:X(\sigma,N_j)\lambdaongrightarrow X^{(j)}\]
be the normalization of $X^{(j)}$ (see Lemma \rhoef{NormTor}). Consider $\nu_j^{-1}(X_{Sing}^{(j)})\sigmaubseteq X(\sigma,N_j)$, it is a disjoint union of orbits
\[\nu_j^{-1}(X_{Sing}^{(j)})=\betaigsqcup_\tau\mbox{orb}(\tau),\ \mbox{ for some }\tau\mbox{ faces of }\sigma.\]
We also have that the complement of $\nu_j^{-1}(X_{Sing}^{(j)})$ in the toric variety $X(\sigma,N_j)$ is a union of orbits.
\betaegin{Defi}
We define, for $j=1,\lambdadots,g$
\[\betaegin{array}{ll}
\sigma_{Sing,j} & \mbox{ the fan associated to }\nu_j^{-1}(X_{Sing}^{(j)})\\
\\
\sigma_{Reg,j} & \mbox{ the fan associated to }X(\sigma,N_j)\sigmaetminus\nu_j^{-1}(X_{Sing}^{(j)})\\
\varepsilonnd{array}\]
For $j=g$ we just denote them by $\sigma_{Sing}$ and $\sigma_{Reg}$ respectively.
\lambdaabel{DefTSigmas}
\varepsilonnd{Defi}
\betaegin{Rem}
Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$, let $\rho_1$ and $\rho_2$ be its one-dimensional faces. For $1\lambdaeq j\lambdaeq g$ we have
\[\sigma_{Sing,j}=\lambdaeft\{\betaegin{array}{cll}
\sigma\sigmaetminus(\rho_1\cup\rho_2) & \mbox{ if } & X_{Sing}^{(j)}=Z_{12}\\
\\
\sigma\sigmaetminus\rho_2 & \mbox{ if } & X_{Sing}^{(j)}=Z_1\\
\\
\sigma\sigmaetminus\{(0,0)\} & \mbox{ if } & X_{Sing}^{(j)}=Z_1\cup Z_2\\
\varepsilonnd{array}\rhoight.\]
and by definition $\sigma_{Reg,j}=\sigma\sigmaetminus\sigma_{Sing,j}$.
\lambdaabel{defSigmas}
\varepsilonnd{Rem}
The fan $\sigma_{Sing}$ will turn out to be necessary in our description of $\pii_m^{-1}(X_{Sing})$ (see Lemma \rhoef{LemGamma}), while the fans $\sigma_{Reg,j}$ will be important in the description of the irreducible components (see Proposition \rhoef{PropC1}).
The sequence
$\{\sigma_{Reg,1},\lambdadots,\sigma_{Reg,g}\}$ is not very complicated, in the sense that most of the elements are the same.
Since by definition $\gammaamma_{g_1+1}^{(2)}=\lambdaambda_{g_1+1}^{(2)}$ then, by Theorem \rhoef{CorSing}, we deduce
\betaegin{equation}
\betaegin{array}{ll}
\mbox{for }1\lambdaeq j\lambdaeq g_2 & \sigmaigma_{Reg,j}=\lambdaeft\{\betaegin{array}{cl}
\rho_1\cup\rho_2 & \mbox{ if }\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }j=1\\
\rho_2 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\\
\\
\mbox{for }g_2+1\lambdaeq j\lambdaeq g & \sigmaigma_{Reg,j}=\{(0,0)\}\\
\varepsilonnd{array}
\lambdaabel{SigRegj}
\varepsilonnd{equation}
Moreover notice that, by definition, we have $\sigma_{Sing,j}\sigmaubseteq\sigma_{Sing,j+1}$. | 4,004 | 114,405 | en |
train | 0.29.5 | \betaegin{Rem}
Notice that
\[\betaegin{array}{l}
Z_1=\{x_1=z=0\}\\
\\
Z_2=\{x_2=f_{g_1}=0\}\\
\\
Z_{12}=\{(0,0,0)\}\\
\varepsilonnd{array}\]
and hence the singular locus of a quasi-ordinary surface singularity is either a point, or a line, or two lines, or a line and a singular curve.
\varepsilonnd{Rem}
Then, geometrically, the meaning of the integer $g_2$ is to measure the irreducibility of the singular locus of the approximated roots. Indeed,
\[\betaegin{array}{l}
X_{Sing}^{(j)} \mbox{ is irreducible, for }1\lambdaeq j\lambdaeq g_2\\
\\
X_{Sing}^{(j)} \mbox{ is reducible, for }g_2<j\lambdaeq g\\
\varepsilonnd{array}\]
Now we define a sequence of semi-open cones keeping track of the singular locus of the quasi-ordinary hypersurfaces $X^{(j)}$ for $j=1,\lambdadots,g$ (see Definition
\rhoef{SemiX}). Let
\[\nu_j:X(\sigma,N_j)\lambdaongrightarrow X^{(j)}\]
be the normalization of $X^{(j)}$ (see Lemma \rhoef{NormTor}). Consider $\nu_j^{-1}(X_{Sing}^{(j)})\sigmaubseteq X(\sigma,N_j)$, it is a disjoint union of orbits
\[\nu_j^{-1}(X_{Sing}^{(j)})=\betaigsqcup_\tau\mbox{orb}(\tau),\ \mbox{ for some }\tau\mbox{ faces of }\sigma.\]
We also have that the complement of $\nu_j^{-1}(X_{Sing}^{(j)})$ in the toric variety $X(\sigma,N_j)$ is a union of orbits.
\betaegin{Defi}
We define, for $j=1,\lambdadots,g$
\[\betaegin{array}{ll}
\sigma_{Sing,j} & \mbox{ the fan associated to }\nu_j^{-1}(X_{Sing}^{(j)})\\
\\
\sigma_{Reg,j} & \mbox{ the fan associated to }X(\sigma,N_j)\sigmaetminus\nu_j^{-1}(X_{Sing}^{(j)})\\
\varepsilonnd{array}\]
For $j=g$ we just denote them by $\sigma_{Sing}$ and $\sigma_{Reg}$ respectively.
\lambdaabel{DefTSigmas}
\varepsilonnd{Defi}
\betaegin{Rem}
Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$, let $\rho_1$ and $\rho_2$ be its one-dimensional faces. For $1\lambdaeq j\lambdaeq g$ we have
\[\sigma_{Sing,j}=\lambdaeft\{\betaegin{array}{cll}
\sigma\sigmaetminus(\rho_1\cup\rho_2) & \mbox{ if } & X_{Sing}^{(j)}=Z_{12}\\
\\
\sigma\sigmaetminus\rho_2 & \mbox{ if } & X_{Sing}^{(j)}=Z_1\\
\\
\sigma\sigmaetminus\{(0,0)\} & \mbox{ if } & X_{Sing}^{(j)}=Z_1\cup Z_2\\
\varepsilonnd{array}\rhoight.\]
and by definition $\sigma_{Reg,j}=\sigma\sigmaetminus\sigma_{Sing,j}$.
\lambdaabel{defSigmas}
\varepsilonnd{Rem}
The fan $\sigma_{Sing}$ will turn out to be necessary in our description of $\pii_m^{-1}(X_{Sing})$ (see Lemma \rhoef{LemGamma}), while the fans $\sigma_{Reg,j}$ will be important in the description of the irreducible components (see Proposition \rhoef{PropC1}).
The sequence
$\{\sigma_{Reg,1},\lambdadots,\sigma_{Reg,g}\}$ is not very complicated, in the sense that most of the elements are the same.
Since by definition $\gammaamma_{g_1+1}^{(2)}=\lambdaambda_{g_1+1}^{(2)}$ then, by Theorem \rhoef{CorSing}, we deduce
\betaegin{equation}
\betaegin{array}{ll}
\mbox{for }1\lambdaeq j\lambdaeq g_2 & \sigmaigma_{Reg,j}=\lambdaeft\{\betaegin{array}{cl}
\rho_1\cup\rho_2 & \mbox{ if }\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }j=1\\
\rho_2 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\\
\\
\mbox{for }g_2+1\lambdaeq j\lambdaeq g & \sigmaigma_{Reg,j}=\{(0,0)\}\\
\varepsilonnd{array}
\lambdaabel{SigRegj}
\varepsilonnd{equation}
Moreover notice that, by definition, we have $\sigma_{Sing,j}\sigmaubseteq\sigma_{Sing,j+1}$.
\betaegin{Defi}
Given $\nu\in\sigma\cap N_0$, we define the following sequence of real numbers
\[l_i(\nu):=n_ie_i\lambdaangle\nu,\gammaamma_i\rhoangle,\mbox{ for }1\lambdaeq i\lambdaeq g.\]
Set $l_0(\nu)=0$ and $l_{g+1}(\nu)=\infty$ for any $\nu\in\mathbb Z^2$.
Moreover, we define
\[i(\nu)=\lambdaeft\{\betaegin{array}{cl}
g+1 & \mbox{ if }\nu\in N_g\\
\\
\mbox{min }\{i\in\{1,\lambdadots,g\}\ |\ \nu\notin N_i\} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
and
\[\betaegin{array}{l}
c(\nu)=\mbox{max }\{0\lambdaeq i\lambdaeq g\}\ |\ \lambdaangle\nu,\gammaamma_i\rhoangle=0\}\\
\\
m(\nu)=\mbox{min }\{1\lambdaeq i\lambdaeq g\ |\ n_i\lambdaangle\nu,\gammaamma_i\rhoangle<\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\}\\
\varepsilonnd{array}\]
\lambdaabel{defs}
\varepsilonnd{Defi}
Notice that, by definition, $l_1(\nu)=l_2(\nu)=\cdots=l_{m(\nu)}(\nu)<l_{m(\nu)+1}(\nu)$. Moreover, for $\nu\in\sigma_{Sing}\cap N_0$, we have that
\[\betaegin{array}{ccc}
c(\nu)=\lambdaeft\{\betaegin{array}{cc}
g_1 & \mbox{ if }g_1>0\mbox{ and }\nu\in\rho_2\\
\\
0 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
&
\mbox{ and }
&
m(\nu)=\lambdaeft\{\betaegin{array}{cc}
\in\{1,\lambdadots,g\} & \mbox{ if }\nu\in\rho_1\\
\\
g_1 & \mbox{ if } \nu\in\rho_2\mbox{ and }g_1>0\\
\\
1 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
\varepsilonnd{array}\]
It is straightforward to check that $c(\nu)\lambdaeq m(\nu)\lambdaeq i(\nu)$.
\betaegin{Lem}
For any $\nu\in\sigma_{Sing}\cap N_0$, we have that
\[l_i(\nu)\in\mathbb Z \mbox{ for }1\lambdaeq i\lambdaeq\mbox{min }\{i(\nu),g\}.\]
Moreover the integers $l_i(\nu)$ are ordered as
\betaegin{equation}
0= l_{c(\nu)}(\nu)< l_{c(\nu)+1}(\nu)\lambdaeq\cdots \lambdaeq l_g(\nu)<l_{g+1}(\nu)=\infty,
\lambdaabel{ineq}
\varepsilonnd{equation}
and we have the equality $l_i(\nu)=l_{i+1}(\nu)$ if and only if $\gammaamma_{i+1}^{(j)}=n_i\gammaamma_i^{(j)}$ for $j$ either $1$ or $2$, and $\nu\in\rho_j$.
For $i>c(\nu)$ we have that the following statements are equivalent:
\betaegin{enumerate}
\item[(i)] $\nu\in N_i$
\item[(ii)] for $1\lambdaeq j\lambdaeq i-1,\ \nu\in N_j$ and the number $\pihirac{l_{j+1}(\nu)-l_j(\nu)}{e_j}$ is a positive integer.
\varepsilonnd{enumerate}
\lambdaabel{l-ord}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
For $1\lambdaeq i<i(\nu)$ we have that $\nu\in N_i$, therefore $\lambdaangle\nu,\gammaamma_i\rhoangle\in\mathbb Z$ and $l_i(\nu)$ is an integer. If $i(\nu)<g+1$, then
$\nu\in N_{i(\nu)-1}$ and by (\rhoef{relgam}) we deduce that $l_{i(\nu)}(\nu)$ is an integer.
By definition of $\gammaamma_i$ it follows that $e_i\gammaamma_{i+1}=e_{i-1}\gammaamma_i+e_i(\lambda_{i+1}-\lambda_i)$. Then
\[l_{i+1}(\nu)=l_i(\nu)+e_i\lambdaangle\nu,\lambda_{i+1}-\lambda_i\rhoangle,\]
and the second claim of the Lemma follows because the exponents $\lambda_i$ are ordered lexicographically as (\rhoef{lex}).
The equivalence follows by the fact that
\[l_j(\nu)-l_{j-1}(\nu)=e_{j-1}(\lambdaangle\nu,\gammaamma_j\rhoangle-n_{j-1}\lambdaangle\nu,\gammaamma_{j-1}\rhoangle).\]
\pisifill $\Box$
\sigmaection{Jet schemes of quasi-ordinary surface singularities}
\lambdaabel{Sec4}
In this section we describe the irreducible components of $\pii_m^{-1}(X_{Sing})\sigmaubset X_m$. We begin with an overview of the section.
We will associate, to any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$, a family of $m$-jets that we call
$C_m^\nu$. Roughly speaking, it is the Zariski closure of the set of $m$-jets whose order of contact with the hyperplane coordinate $x_i$ is bigger or equal to $\nu_i$, for $i=1,2$.
We divide the sets $C_m^\nu$ into two types. Sets defined by the annihilation of hyperplane coordinates in $\mathbb A_m^3=
\mbox{Spec }\mathbb C[x_1^{(i)},x_2^{(i)},z^{(i)}]_{i=0,\lambdadots,m}$, which are associated with $\nu$ in a certain set $H_m\sigmaubset\sigmaigma_{Sing}\cap N_0$ (where $H$ stands for hyperplane). The $C_m^\nu$ of the second type have a more complicated geometry and are associated with $\nu$ in a certain set $L_m$ (where $L$ stands for
lattice, because for such components, $\nu$ belongs to one of the lattices $N_i$ for $1\lambdaeq i\lambdaeq g$).
The geometry of $C_m^\nu$ for $\nu\in H_m$ is evident and the possible inclusions among different $C_m^\nu$ are determined only by looking at the
product ordering of the associated vectors $\nu$. For $\nu\in L_m$, to understand the geometry of $C_m^\nu$, we consider the order of contact of the generic point
of $C_m^\nu$ with the different approximated roots. This allows us to detect certain dense subset of $C_m^\nu$, which is isomorphic to the cartesian product of an open set of the spectrum of the graded algebra (associated with the last approximated root appearing in the equations defining $C_m^{\nu}$) and an affine
space. This permits to prove that each $C_m^\nu$ is irreducible and to compute its codimension (see Proposition \rhoef{Prop1}).
The inclusions among the $C_m^\nu$ for $\nu\in H_m\cup L_m$ are more delicate. We introduce in Definition \rhoef{orden2} a new relation to detect such
inclusions. In this definition, the singular locus of the last approximated root affecting the geometry of $C_m^\nu$, plays a crucial role.
Finally, with the collection of sets $C_m^\nu$ left, we prove in Theorem \rhoef{TheCaso1} that they are the irreducible components of the $m$-jets
through the singular locus.
\betaegin{Defi}
Let $h\in \mathbb C[x_1,\lambdadots,x_n]$ and let $X$ be an algebraic variety. For $p,m\in\mathbb Z_{>0}$ with $p\lambdaeq m$ we set
\[\mbox{Cont}_X^p(h)_m:=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(h\circ\gammaamma)=p\}.\]
And, for $m\in\mathbb Z_{>0}$ and any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $\nu_i\lambdaeq m$ we define the constructible set
\[D_m^\nu=(\mbox{Cont}_X^{\nu_1}(x_1)_m\cap\mbox{Cont}_X^{\nu_2}(x_2)_m)_{red},\]
where we consider the reduced structure, and
\[C_m^\nu=\mathbb overline{D_m^\nu},\]
its Zariski closure. We denote by $D(f)$ the open set
\[D(f)=\mbox{Spec }R_f\]
where $R$ is the ring $R=\mathbb C[x_1^{(j)},x_2^{(j)},z^{(j)}]_{j\gammaeq 0}$.
\lambdaabel{DefDC}
\varepsilonnd{Defi}
Given a jet $\gammaamma\in X_m$, if $x_i\circ\gammaamma\neq0$ for $i=1,2$, the vector $\nu=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))$ belongs to $\sigma\cap N_0$ and $0\lambdaeq\nu_i\lambdaeq m$. Moreover it is trivial that $\gammaamma\in C_m^\nu$. Now we look at $m$-jets with origin at the singular locus.
\betaegin{Lem}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, there exists $\nu\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$, such that $\gammaamma\in C_m^\nu$.
Moreover
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu.\]
\lambdaabel{LemGamma}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, suppose first that $x_i\circ\gammaamma\neq 0$ for $i=1,2$. Then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and obviously $\gammaamma\in D_m^\nu\sigmaubseteq C_m^\nu$. We have to prove that $\nu\in\sigma_{Sing}\cap N_0$, and this follows by Remark \rhoef{defSigmas}, since:
\betaegin{enumerate}
\item[(i)] If $X_{Sing}=\{(0,0,0)\}$, then $\gammaamma(0)=(0,0,0)$ and ord$_t(x_i\circ\gammaamma)>0$ for $i=1,2$. | 4,063 | 114,405 | en |
train | 0.29.6 | We will associate, to any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$, a family of $m$-jets that we call
$C_m^\nu$. Roughly speaking, it is the Zariski closure of the set of $m$-jets whose order of contact with the hyperplane coordinate $x_i$ is bigger or equal to $\nu_i$, for $i=1,2$.
We divide the sets $C_m^\nu$ into two types. Sets defined by the annihilation of hyperplane coordinates in $\mathbb A_m^3=
\mbox{Spec }\mathbb C[x_1^{(i)},x_2^{(i)},z^{(i)}]_{i=0,\lambdadots,m}$, which are associated with $\nu$ in a certain set $H_m\sigmaubset\sigmaigma_{Sing}\cap N_0$ (where $H$ stands for hyperplane). The $C_m^\nu$ of the second type have a more complicated geometry and are associated with $\nu$ in a certain set $L_m$ (where $L$ stands for
lattice, because for such components, $\nu$ belongs to one of the lattices $N_i$ for $1\lambdaeq i\lambdaeq g$).
The geometry of $C_m^\nu$ for $\nu\in H_m$ is evident and the possible inclusions among different $C_m^\nu$ are determined only by looking at the
product ordering of the associated vectors $\nu$. For $\nu\in L_m$, to understand the geometry of $C_m^\nu$, we consider the order of contact of the generic point
of $C_m^\nu$ with the different approximated roots. This allows us to detect certain dense subset of $C_m^\nu$, which is isomorphic to the cartesian product of an open set of the spectrum of the graded algebra (associated with the last approximated root appearing in the equations defining $C_m^{\nu}$) and an affine
space. This permits to prove that each $C_m^\nu$ is irreducible and to compute its codimension (see Proposition \rhoef{Prop1}).
The inclusions among the $C_m^\nu$ for $\nu\in H_m\cup L_m$ are more delicate. We introduce in Definition \rhoef{orden2} a new relation to detect such
inclusions. In this definition, the singular locus of the last approximated root affecting the geometry of $C_m^\nu$, plays a crucial role.
Finally, with the collection of sets $C_m^\nu$ left, we prove in Theorem \rhoef{TheCaso1} that they are the irreducible components of the $m$-jets
through the singular locus.
\betaegin{Defi}
Let $h\in \mathbb C[x_1,\lambdadots,x_n]$ and let $X$ be an algebraic variety. For $p,m\in\mathbb Z_{>0}$ with $p\lambdaeq m$ we set
\[\mbox{Cont}_X^p(h)_m:=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(h\circ\gammaamma)=p\}.\]
And, for $m\in\mathbb Z_{>0}$ and any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $\nu_i\lambdaeq m$ we define the constructible set
\[D_m^\nu=(\mbox{Cont}_X^{\nu_1}(x_1)_m\cap\mbox{Cont}_X^{\nu_2}(x_2)_m)_{red},\]
where we consider the reduced structure, and
\[C_m^\nu=\mathbb overline{D_m^\nu},\]
its Zariski closure. We denote by $D(f)$ the open set
\[D(f)=\mbox{Spec }R_f\]
where $R$ is the ring $R=\mathbb C[x_1^{(j)},x_2^{(j)},z^{(j)}]_{j\gammaeq 0}$.
\lambdaabel{DefDC}
\varepsilonnd{Defi}
Given a jet $\gammaamma\in X_m$, if $x_i\circ\gammaamma\neq0$ for $i=1,2$, the vector $\nu=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))$ belongs to $\sigma\cap N_0$ and $0\lambdaeq\nu_i\lambdaeq m$. Moreover it is trivial that $\gammaamma\in C_m^\nu$. Now we look at $m$-jets with origin at the singular locus.
\betaegin{Lem}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, there exists $\nu\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$, such that $\gammaamma\in C_m^\nu$.
Moreover
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu.\]
\lambdaabel{LemGamma}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, suppose first that $x_i\circ\gammaamma\neq 0$ for $i=1,2$. Then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and obviously $\gammaamma\in D_m^\nu\sigmaubseteq C_m^\nu$. We have to prove that $\nu\in\sigma_{Sing}\cap N_0$, and this follows by Remark \rhoef{defSigmas}, since:
\betaegin{enumerate}
\item[(i)] If $X_{Sing}=\{(0,0,0)\}$, then $\gammaamma(0)=(0,0,0)$ and ord$_t(x_i\circ\gammaamma)>0$ for $i=1,2$.
\item[(ii)] If $X_{Sing}=Z_1$, then $\gammaamma(0)=(0,x_2(0),0)$, and $\mbox{ord}_t(x_1\circ\gammaamma)>0,\ \mbox{ord}_t(x_2\circ\gammaamma)\gammaeq 0$.
\item[(iii)] If $X_{Sing}=Z_1\cup Z_2$, then
\[\mbox{if }\gammaamma(0)\in Z_1, \mbox{ we have ord}_t(x_1\circ\gammaamma)>0\mbox{ and ord}_t(x_2\circ\gammaamma)\gammaeq 0\]
\[\mbox{if }\gammaamma(0)\in Z_2, \mbox{ we have ord}_t(x_1\circ\gammaamma)\gammaeq 0\mbox{ and ord}_t(x_2\circ\gammaamma)>0\]
\varepsilonnd{enumerate}
To deal with the other cases, notice that
\[C_m^\nu=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(x_i\circ\gammaamma)\gammaeq\nu_i,\ i=1,2\}.\]
If $x_i\circ\gammaamma=0$ for $i=1,2$, then $\gammaamma\in C_m^\nu$ for any $\nu\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$.
If $x_1\circ\gammaamma=0$ and $x_2\circ\gammaamma\neq 0$, then we denote $\alphalpha:=\mbox{ord}_t(x_2\circ\gammaamma)$. We have $0\lambdaeq\alphalpha\lambdaeq m$, and $\gammaamma\in C_m^\nu$
for any $\nu\in\sigma_{Sing}\cap N_0$, with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$, and $\nu_2\lambdaeq\alphalpha$.
The left case $x_1\circ\gammaamma\neq 0$ and $x_2\circ\gammaamma=0$ is analogous to the last one.
We prove the other inclusion. If $\gammaamma\in X_m\sigmaetminus\pii_m^{-1}(X_{Sing})$, then $\gammaamma(0)\notin X_{Sing}$. Again distinguishing cases depending on the singular locus, we can prove that $\nu=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\notin\sigma_{Sing}$.
\pisifill$\Box$
\sigmaubsection{Description of the sets $C_m^\nu$}
The sets $C_m^\nu$ are the candidates to be irreducible components of $\pii_m^{-1}(X_{Sing})$. We proceed to study these sets. Notice that, by definition, it follows that
\[C_m^\nu\sigmaubset V(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)},i=1,2).\]
\betaegin{Defi}
For $\nu\in\sigma\cap N_0$ and $m\in\mathbb Z_{>0}$ we define the ideals
\[I^\nu=\lambdaeft(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)}\rhoight)_{i=1,2},\]
\[J_m^\nu=Rad\lambdaeft((F^{(i)}\mbox{ mod }I^\nu)_{0\lambdaeq i\lambdaeq m}\rhoight).\]
Moreover we define the integers $j(m,\nu)\in\{0,\lambdadots,i(\nu)-1\}$ and $j'(m,\nu)\in\{0,\lambdadots,j(m,\nu)\}$ by the
inequalities
\[\betaegin{array}{c}
l_j(\nu)\lambdaeq m< l_{j+1}(\nu),\\
\\
l_j(\nu)+e_j\lambdaeq m<l_{j+1}(\nu)+e_{j+1},\\
\varepsilonnd{array}\]
respectively.
\lambdaabel{IJ}
\varepsilonnd{Defi}
Then we have that
\betaegin{equation}
D_m^\nu=V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)}),
\lambdaabel{eqast}
\varepsilonnd{equation}
where the fact of taking the radical in the definition of $J_m^\nu$ corresponds to taking the reduced structure in the definition of $D_m^\nu$.
We have to study the polynomials $F^{(j)}\mbox{ mod }I^\nu$ for $0\lambdaeq j\lambdaeq m$. Thanks to the identification in (\rhoef{Desc1}), these polynomials have to be seen as the defining equations of $D_m^\nu$.
\betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The singular locus of $X$ is reducible
\[X_{Sing}=\{x_1=z=0\}\cup\{x_2=z=0\}.\]
We described in Example \rhoef{Ex0} the equations of the $3$-jets, and if we look at $3$-jets with origin in the singular locus, then we have to add the condition either $x_1^{(0)}=z^{(0)}=0$ or $x_2^{(0)}=z^{(0)}=0$. This is equivalent to consider the equations modulo the ideal $I^\nu$.
Then
\[\betaegin{array}{lcl}
(\pii_1^{-1}(X_{Sing}))_{red} & = & \pii_1^{-1}(\{x_1=z=0\})\cup\pii_1^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},z^{(0)})=C_1^{(1,0)}\cup C_1^{(0,1)}\sigmaubset\mathbb A_1^3.\\
\\
(\pii_2^{-1}(X_{Sing}))_{red} & = & \pii_2^{-1}(\{x_1=z=0\})\cup\pii_2^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup (V(x_2^{(0)},z^{(0)},{x_1^{(0)}}^3{x_2^{(1)}}^2))_{red}=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_1^{(0)},x_2^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=C_2^{(1,0)}\cup C_2^{(0,2)}\sigmaubset\mathbb A_2^3,\\
\varepsilonnd{array}\]
since $V(x_1^{(0)},x_2^{(0)},z^{(0)})\sigmaubset V(x_1^{(0)},z^{(0)})$. And
\[(\pii_3^{-1}(X_{Sing}))_{red}=V(x_1^{(0)},z^{(0)},{z^{(1)}}^3-{x_1^{(1)}}^3{x_2^{(0)}}^2)\cup V(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)})=C_3^{(1,0)}\cup C_3^{(0,2)}
\sigmaubset\mathbb A_3^3.\]
\lambdaabel{Ex1}
\varepsilonnd{Exam}
In this example we see how the components are defined by hyperplane coordinates for $m<3$, and at level $m=3$ the equation $f$ starts playing a role. When there are more than one approximated root, the approximated roots affect the geometry of $C_m^\nu$ one after the other as $m$ grows. This will be explained in Proposition \rhoef{Cgeom}. We illustrate this with another example.
\betaegin{Exam}
Consider the quasi-ordinary surface $f=(z^2-x_1^3)^3-x_1^{10}x_2^4$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},0)$ and $\gammaamma_2=(\pihirac{10}{3},
\pihirac{4}{3})$, and the singular locus is $X_{Sing}=\{x_1=z=0\}\cup\{x_2=z^2-x_1^3=0\}$.
If we lift the component of the singular locus $Z_2=\{x_2=f_1=0\}$ at level $3$, we have that
$(\pii_3^{-1}(Z_2))_{red}=V(x_2^{(0)},F_1^{(0)},F_2^{(1)},F_2^{(2)},F_2^{(3)})$, where $F_1^{(0)}={z^{(0)}}^2-{x_1^{(0)}}^3$, and
\[\betaegin{array}{ll}
F_2^{(1)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(1)}\mbox{ mod }(x_2^{(0)},z^{(0)})\\
& \varepsilonquiv 0\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)})\\
\\
F_2^{(2)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(2)}+6F_1^{(0)}{F_1^{(1)}}^2\mbox{ mod }(x_2^{(0)},z^{(0)})\\ | 3,903 | 114,405 | en |
train | 0.29.7 | Then we have that
\betaegin{equation}
D_m^\nu=V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)}),
\lambdaabel{eqast}
\varepsilonnd{equation}
where the fact of taking the radical in the definition of $J_m^\nu$ corresponds to taking the reduced structure in the definition of $D_m^\nu$.
We have to study the polynomials $F^{(j)}\mbox{ mod }I^\nu$ for $0\lambdaeq j\lambdaeq m$. Thanks to the identification in (\rhoef{Desc1}), these polynomials have to be seen as the defining equations of $D_m^\nu$.
\betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The singular locus of $X$ is reducible
\[X_{Sing}=\{x_1=z=0\}\cup\{x_2=z=0\}.\]
We described in Example \rhoef{Ex0} the equations of the $3$-jets, and if we look at $3$-jets with origin in the singular locus, then we have to add the condition either $x_1^{(0)}=z^{(0)}=0$ or $x_2^{(0)}=z^{(0)}=0$. This is equivalent to consider the equations modulo the ideal $I^\nu$.
Then
\[\betaegin{array}{lcl}
(\pii_1^{-1}(X_{Sing}))_{red} & = & \pii_1^{-1}(\{x_1=z=0\})\cup\pii_1^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},z^{(0)})=C_1^{(1,0)}\cup C_1^{(0,1)}\sigmaubset\mathbb A_1^3.\\
\\
(\pii_2^{-1}(X_{Sing}))_{red} & = & \pii_2^{-1}(\{x_1=z=0\})\cup\pii_2^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup (V(x_2^{(0)},z^{(0)},{x_1^{(0)}}^3{x_2^{(1)}}^2))_{red}=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_1^{(0)},x_2^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=C_2^{(1,0)}\cup C_2^{(0,2)}\sigmaubset\mathbb A_2^3,\\
\varepsilonnd{array}\]
since $V(x_1^{(0)},x_2^{(0)},z^{(0)})\sigmaubset V(x_1^{(0)},z^{(0)})$. And
\[(\pii_3^{-1}(X_{Sing}))_{red}=V(x_1^{(0)},z^{(0)},{z^{(1)}}^3-{x_1^{(1)}}^3{x_2^{(0)}}^2)\cup V(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)})=C_3^{(1,0)}\cup C_3^{(0,2)}
\sigmaubset\mathbb A_3^3.\]
\lambdaabel{Ex1}
\varepsilonnd{Exam}
In this example we see how the components are defined by hyperplane coordinates for $m<3$, and at level $m=3$ the equation $f$ starts playing a role. When there are more than one approximated root, the approximated roots affect the geometry of $C_m^\nu$ one after the other as $m$ grows. This will be explained in Proposition \rhoef{Cgeom}. We illustrate this with another example.
\betaegin{Exam}
Consider the quasi-ordinary surface $f=(z^2-x_1^3)^3-x_1^{10}x_2^4$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},0)$ and $\gammaamma_2=(\pihirac{10}{3},
\pihirac{4}{3})$, and the singular locus is $X_{Sing}=\{x_1=z=0\}\cup\{x_2=z^2-x_1^3=0\}$.
If we lift the component of the singular locus $Z_2=\{x_2=f_1=0\}$ at level $3$, we have that
$(\pii_3^{-1}(Z_2))_{red}=V(x_2^{(0)},F_1^{(0)},F_2^{(1)},F_2^{(2)},F_2^{(3)})$, where $F_1^{(0)}={z^{(0)}}^2-{x_1^{(0)}}^3$, and
\[\betaegin{array}{ll}
F_2^{(1)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(1)}\mbox{ mod }(x_2^{(0)},z^{(0)})\\
& \varepsilonquiv 0\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)})\\
\\
F_2^{(2)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(2)}+6F_1^{(0)}{F_1^{(1)}}^2\mbox{ mod }(x_2^{(0)},z^{(0)})\\
& \varepsilonquiv 0\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)})\\
\\
F_2^{(3)} & \varepsilonquiv {F_1^{(1)}}^3\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)}),\\
\varepsilonnd{array}\]
and then $(\pii_3^{-1}(Z_2))_{red}=V(x_2^{(0)},F_1^{(0)},F_1^{(1)})$. Notice that it is not a component of $(\pii_3^{-1}(X_{Sing}))_{red},$ since it is not irreducible. Indeed, it decomposes as
\[(\pii_3^{-1}(Z_2))_{red}=V(x_1^{(0)},x_2^{(0)},z^{(0)})\cup\mathbb overline{V(x_2^{(0)},F_1^{(0)},F_1^{(1)})\cap D(x_1^{(0)})}.\]
\lambdaabel{Ex2}
\varepsilonnd{Exam}
We have seen in this example how, to give a minimal set of generators of $J_m^\nu$, we need to study the polynomials $F^{(l)}\mbox{ mod }J_{l-1}^\nu$.
Therefore we introduce the following definition.
\betaegin{Defi}
For $\nu\in \sigma_{Sing}\cap N_0$ and $0\lambdaeq l\lambdaeq m$, we denote by
\[F_{\nu}^{(l)}:=F^{(l)}\mbox{ mod }(I^\nu,J_{l-1}^\nu),\]
and for the approximated roots the notation is $F_{j,\nu}^{(l)}$.
\lambdaabel{Fnu}
\varepsilonnd{Defi}
And now we obviously have that
\[J_m^\nu=Rad(F_\nu^{(0)},\lambdadots,F_\nu^{(m)}).\]
Regarding the claim in Remark \rhoef{RemTonto}, once we consider $F_\nu^{(l)}$, it is not true anymore that the polynomials are non-zero. But, whenever $F_\nu^{(l)}$ is non-zero, then it is quasi-homogeneous of degree $l$.
In general, the first approximated root which appears is not necessarily the first one, and the process does not finish with the last one.
To control, for a given $\nu$, all this behaviour, we defined the integers $i(\nu),c(\nu)$ and $m(\nu)$ in Definition \rhoef{defs}.
Indeed, given $m\in\mathbb Z_{>0}$ and $\nu\in\sigma_{Sing}\cap N_0$ such that $l_{c(\nu)}(\nu)\lambdaeq m$, the approximated roots which will influence the defining ideal of $D_m^\nu$, are
\[f_{c(\nu)},\lambdadots,f_{j(m,\nu)},\]
where remember the convention $f_0=z$. Moreover, the moment when $f_i$ begins to influence the defining equations of $C_m^\nu$ (or in other words, the generators of $J_m^\nu$) for the first time is exactly at $m=l_i(\nu)$. This is the content of Corollary \rhoef{Corolario3}.
The meaning of the integer $i(\nu)$ is that, at $m=l_{i(\nu)}(\nu)$, $\nu$ does no longer give rise to an irreducible component (see Lemma \rhoef{LemD}).
The integer $j(m,\nu)$ will be useful to describe the component $C_m^\nu$ (see Proposition \rhoef{Cgeom}), while $j'(m,\nu)$ will be crucial when studying the inclusion $C_m^{\nu'}\sigmaubseteq C_m^\nu$ (see Proposition \rhoef{PropC1}).
\betaegin{Exam}
We revisit Example \rhoef{Ex1}. If we lift the component $C_3^{(0,2)}$ to level $4$, we have
\[\pii_{4,3}^{-1}(C_3^{(0,2)})=V(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)},F^{(4)})\]
where $F^{(4)}\varepsilonquiv {x_1^{(0)}}^3{x_2^{(2)}}^2\mbox{ mod }(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)})$. Therefore
\[\pii_{4,3}^{-1}(C_3^{(0,2)})=C_4^{(1,2)}\cup C_4^{(0,3)}.\]
Then at level $m=4$, the vector $(0,2)$ does not give rise to an irreducible component any longer. The reason is that $(0,2)\notin N_1$ and $4=l_1(0,2)$.
\lambdaabel{Ex1re}
\varepsilonnd{Exam}
This is the case in general as we claim in the next Lemma, whose proof is left to Section \rhoef{Proofs}.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$, we have
\[D_m^\nu=\varepsilonmptyset\mbox{ if and only if }m\gammaeq l_{i(\nu)}(\nu).\]
\lambdaabel{LemD}
\varepsilonnd{Lem}
As a consequence of this Lemma, we are going to prove an improvement of Lemma \rhoef{LemGamma}, namely, for $m\in\mathbb Z_{>0}$, to cover $\pii_m^{-1}(X_{Sing})$ it is enough to consider $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$ with $m<l_{i(\nu)}(\nu)$.
Notice that, if $l_{m(\nu)}(\nu)\lambdaeq m$, then $l_1(\nu)\lambdaeq l_{m(\nu)}(\nu)\lambdaeq m$, and by the previous Lemma, we have to ask $\nu\in N_1$ whenever $l_{m(\nu)}\lambdaeq m$.
\betaegin{Defi}
Given $m\in\mathbb Z_{> 0}$ we define the sets:
\[\betaegin{array}{l}
H_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0\ |\ l_{m(\nu)}(\nu)\gammaeq m+1\},\\
\\
L_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_1\ |\ l_{m(\nu)}(\nu)\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
It will be necessary later to subdivide the set $L_m$ as
\[\betaegin{array}{l}
L_m^==\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)\lambdaeq m<\mbox{min}\{l_{m(\nu)}(\nu)+e_{m(\nu)},l_{i(\nu)}(\nu)\}\},\\
\\
L_m^<=\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)+e_{m(\nu)}\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
\lambdaabel{DefsSets}
\varepsilonnd{Defi}
If we come back to Example \rhoef{Ex1}, we have that
\[\betaegin{array}{ll}
(\pii_1^{-1}(X_{Sing}))_{red}=C_1^{(1,0)}\cup C_1^{(0,1)} & \mbox{ with }(1,0),(0,1)\in H_1,\\
\\
(\pii_2^{-1}(X_{Sing}))_{red}=C_2^{(1,0)}\cup C_2^{(0,2)} & \mbox{ with }(1,0),(0,2)\in H_2,\\
\\
(\pii_3^{-1}(X_{Sing}))_{red}=C_3^{(1,0)}\cup C_3^{(0,2)} & \mbox{ with }(1,0)\in L_3^=\mbox{ and }(0,2)\in H_3.\\
\varepsilonnd{array}\]
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$, we have that $H_m\cup L_m\neq \varepsilonmptyset$, and
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu.\]
\lambdaabel{LemTh}
\varepsilonnd{Lem}
{\varepsilonm Proof.} The first claim follows because $(m,m)\in H_m$ for any $m\in\mathbb Z_{>0}$. Indeed, since $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=e_1m(a_1+b_1)>m$, where the last inequality holds because $a_1+b_1>1$, since the branch is normalized.
By Lemma \rhoef{LemGamma}
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu\sigmaubseteq\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu=\pii_m^{-1}(X_{Sing}).\]
We prove the other inclusion. Notice that $\nu\notin H_m\cup L_m$ is equivalent to $l_{i(\nu)}(\nu)\lambdaeq m$, because $m(\nu)\lambdaeq i(\nu)$. For any $\gammaamma\in\pii_m^{-1}(X_{Sing})$,
$\betaullet$ if $x_i\circ\gammaamma\neq 0$ for $i=1,2$, then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and $\gammaamma\in D_m^\nu$. Hence, by Lemma \rhoef{LemD} we have that $m<l_{i(\nu)}(\nu)$ and therefore $\nu\in H_m\cup L_m$.
Otherwise,
$\betaullet$ if $x_i\circ\gammaamma=0$ for $i=1,2$ we saw in the proof of Lemma \rhoef{LemGamma} that $\gammaamma\in C_m^\nu$ for any $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$. Therefore, since $H_m\cap L_m\neq\varepsilonmptyset$, there exists $\nu$ with $\gammaamma\in C_m^\nu$. | 4,010 | 114,405 | en |
train | 0.29.8 | This is the case in general as we claim in the next Lemma, whose proof is left to Section \rhoef{Proofs}.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$, we have
\[D_m^\nu=\varepsilonmptyset\mbox{ if and only if }m\gammaeq l_{i(\nu)}(\nu).\]
\lambdaabel{LemD}
\varepsilonnd{Lem}
As a consequence of this Lemma, we are going to prove an improvement of Lemma \rhoef{LemGamma}, namely, for $m\in\mathbb Z_{>0}$, to cover $\pii_m^{-1}(X_{Sing})$ it is enough to consider $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$ with $m<l_{i(\nu)}(\nu)$.
Notice that, if $l_{m(\nu)}(\nu)\lambdaeq m$, then $l_1(\nu)\lambdaeq l_{m(\nu)}(\nu)\lambdaeq m$, and by the previous Lemma, we have to ask $\nu\in N_1$ whenever $l_{m(\nu)}\lambdaeq m$.
\betaegin{Defi}
Given $m\in\mathbb Z_{> 0}$ we define the sets:
\[\betaegin{array}{l}
H_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0\ |\ l_{m(\nu)}(\nu)\gammaeq m+1\},\\
\\
L_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_1\ |\ l_{m(\nu)}(\nu)\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
It will be necessary later to subdivide the set $L_m$ as
\[\betaegin{array}{l}
L_m^==\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)\lambdaeq m<\mbox{min}\{l_{m(\nu)}(\nu)+e_{m(\nu)},l_{i(\nu)}(\nu)\}\},\\
\\
L_m^<=\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)+e_{m(\nu)}\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
\lambdaabel{DefsSets}
\varepsilonnd{Defi}
If we come back to Example \rhoef{Ex1}, we have that
\[\betaegin{array}{ll}
(\pii_1^{-1}(X_{Sing}))_{red}=C_1^{(1,0)}\cup C_1^{(0,1)} & \mbox{ with }(1,0),(0,1)\in H_1,\\
\\
(\pii_2^{-1}(X_{Sing}))_{red}=C_2^{(1,0)}\cup C_2^{(0,2)} & \mbox{ with }(1,0),(0,2)\in H_2,\\
\\
(\pii_3^{-1}(X_{Sing}))_{red}=C_3^{(1,0)}\cup C_3^{(0,2)} & \mbox{ with }(1,0)\in L_3^=\mbox{ and }(0,2)\in H_3.\\
\varepsilonnd{array}\]
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$, we have that $H_m\cup L_m\neq \varepsilonmptyset$, and
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu.\]
\lambdaabel{LemTh}
\varepsilonnd{Lem}
{\varepsilonm Proof.} The first claim follows because $(m,m)\in H_m$ for any $m\in\mathbb Z_{>0}$. Indeed, since $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=e_1m(a_1+b_1)>m$, where the last inequality holds because $a_1+b_1>1$, since the branch is normalized.
By Lemma \rhoef{LemGamma}
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu\sigmaubseteq\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu=\pii_m^{-1}(X_{Sing}).\]
We prove the other inclusion. Notice that $\nu\notin H_m\cup L_m$ is equivalent to $l_{i(\nu)}(\nu)\lambdaeq m$, because $m(\nu)\lambdaeq i(\nu)$. For any $\gammaamma\in\pii_m^{-1}(X_{Sing})$,
$\betaullet$ if $x_i\circ\gammaamma\neq 0$ for $i=1,2$, then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and $\gammaamma\in D_m^\nu$. Hence, by Lemma \rhoef{LemD} we have that $m<l_{i(\nu)}(\nu)$ and therefore $\nu\in H_m\cup L_m$.
Otherwise,
$\betaullet$ if $x_i\circ\gammaamma=0$ for $i=1,2$ we saw in the proof of Lemma \rhoef{LemGamma} that $\gammaamma\in C_m^\nu$ for any $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$. Therefore, since $H_m\cap L_m\neq\varepsilonmptyset$, there exists $\nu$ with $\gammaamma\in C_m^\nu$.
$\betaullet$ if $x_1\circ\gammaamma=0$ and $x_2\circ\gammaamma\neq 0$, then by the proof of Lemma \rhoef{LemGamma}, we have that $\gammaamma\in C_m^{(m,\alphalpha)}$, where $\alphalpha=\mbox{ord}_t(x_2\circ\gammaamma)$. We have to prove that $\nu:=(m,\alphalpha)\in H_m$, and this follows since $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=e_1(a_1m+b_1\alphalpha)>m$, again using that the branch is normalized.
$\betaullet$ if $x_1\circ\gammaamma\neq 0$ and $x_2\circ\gammaamma=0$, then by the proof of Lemma \rhoef{LemGamma}, $\gammaamma\in C_m^\nu$ for any $\nu$ with $\nu_1\lambdaeq\alphalpha$ and $\nu_2\lambdaeq m$, where $\alphalpha=\mbox{ord}_t(x_1\circ\gammaamma)$. Hence we only have to prove that $([0,\alphalpha]\tauimes[0,m])\cap(H_m\cup L_m)\neq\varepsilonmptyset$. If $b_1\gammaeq 1$ then $\nu:=(\alphalpha,m)\in H_m$. Indeed, if $b_1>1$ clearly $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=e_1(a_1\alphalpha+b_1m)>m$. The same works if $b_1=1$ and $g>1$, because then $e_1>1$. If $b_1=1$ and $g=1$, then by Theorem \rhoef{CorSing} we deduce that $\rho_2\nsubseteq\sigma_{Sing}$. Then $\alphalpha>0$ and it follows that $\nu\in H_m$.
The case left is $b_1=0$. In this case, if we set $\nu:=(\alphalpha,m)$, we have $m(\nu)=g_1$ and $i(\nu)\gammaeq g_1+1$. Then $l_{i(\nu)}(\nu)\gammaeq l_{g_1+1}(\nu)=e_{g_1+1}(\alphalpha n_{g_1+1}\gammaamma_{g_1+1}^{(1)}+mn_{g_1+1}\gammaamma_{g_1+1}^{(2)})>m$, where we are using that if $\alphalpha=0$ then $\rho_2\sigmaubseteq\sigma_{Sing}$ and $n_{g_1+1}\gammaamma_{g_1+1}^{(2)}>1$. If $\alphalpha\gammaeq n_1$, the same argument shows that $(n_1,m)\in L_m$. Otherwise $(\alphalpha,m)\in L_m$, since $\gammaamma\in X_m$ and $\gammaamma_1=(\pihirac{a_1}{n_1},0)$, therefore $a_1\mbox{ord}_t(x_1\circ\gammaamma)=n_1\mbox{ord}_t(z\circ\gammaamma)$, which implies that $\alphalpha\pihirac{a_1}{n_1}\in\mathbb Z$ or in other words, $(\alphalpha,m)\in N_1$.
\pisifill $\Box$
Given $\nu\in\sigma_{Sing}\cap N_0$, it gives rise to a candidate of irreducible component at level $m$, $C_m^\nu$, for
\[0=l_{c(\nu)}(\nu)\lambdaeq m<l_{i(\nu)}(\nu).\]
\betaegin{Rem}
\betaegin{enumerate}
\item[(i)] If $\nu\in H_m$, then $c(\nu)=0$, since otherwise $m(\nu)=c(\nu)$ and $l_{m(\nu)}(\nu)=0<m+1$, or in other words, if $g_1>0$ then $H_m\cap\rho_2=\varepsilonmptyset$.
\item[(ii)] It is clear that
$j(m,\nu)=0 \mbox{ if and only if }\nu\in H_m$ and $j(m,\nu)\gammaeq 1 \mbox{ if and only if }\nu\in L_m$.
\varepsilonnd{enumerate}
\lambdaabel{RemDefHL}
\varepsilonnd{Rem}
For $\nu\in H_m$ the sets $C_m^\nu$ are very easy to describe, as we see in the next Proposition.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m$ we have that
\[J_m^\nu=(z^{(0)},\lambdadots,z^{([m/n])}),\]
and hence the set $C_m^\nu$ is defined by hyperplane coordinates in $\mathbb A_m^3$ as
\[C_m^\nu=V(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)},i=1,2;z^{(0)},\lambdadots,z^{([\pihirac{m}{n}])}).\]
\lambdaabel{CHm}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
The proof is by induction on $m$. For $m=1$ we have
\[H_1=\lambdaeft\{\betaegin{array}{ll}
\{(1,1)\} & \mbox{ if }\gammaamma_1=(\pihirac{1}{n},\pihirac{1}{n})\mbox{ and }g=1\\
\\
\{(1,0),(1,1)\} & \mbox{ if }g_1>0\mbox{ (recall that the branch is normalized)}\\
\\
\{(1,0),(1,1),(0,1)\} &\mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
and the claim follows, since
\[C_1^{(1,0)}=V(x_1^{(0)},z^{(0)}),\ C_1^{(1,1)}=V(x_1^{(0)},x_2^{(0)},z^{(0)})\mbox{ and }\ C_1^{(0,1)}=V(x_2^{(0)},z^{(0)})\mbox{ if }g_1=0.\]
Suppose the claim is true for $m$ and we will prove it for $m+1$. Given $\nu\in H_{m+1}$, since $\nu\in H_m$, by induction hypothesis we have that
\[J_{m+1}^\nu=(z^{(0)},\lambdadots,z^{([m/n])},F_\nu^{(m+1)}).\]
We claim that
\[F_\nu^{(m+1)}=\lambdaeft\{\betaegin{array}{cl}
0 & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
{z^{(\pihirac{m+1}{n})}}^n & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
which proves the result, since
\[\lambdaeft[\pihirac{m+1}{n}\rhoight]=\lambdaeft\{\betaegin{array}{cl}
[\pihirac{m}{n}] & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
\pihirac{m+1}{n} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
If $F_\nu^{(m+1)}\neq 0$ then it is a quasi-homogeneous polynomial of degree $m+1$. By the expansion (\rhoef{grgr1}) given in Lemma \rhoef{LemExpSR}
\[f=z^n+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1>n_1e_1\gammaamma_1=n\gammaamma_1$. Then any monomial in $f-z^n$ verifies for any $\gammaamma\in D_m^\nu$
\[\betaegin{array}{ll}
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\mbox{ord}_t(z\circ\gammaamma)\\
& \gammaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\pihirac{m+1}{n}\\
& \gammaeq n\lambdaangle\nu,\gammaamma_1\rhoangle-r_1\lambdaangle\nu,\gammaamma_1\rhoangle+r_1\pihirac{m+1}{n}\\
& =(n-r_1)\pihirac{l_1(\nu)}{n}+r_1\pihirac{m+1}{n}>(n-r_1)\pihirac{m+1}{n}+r_1\pihirac{m+1}{n}=m+1\\
\varepsilonnd{array}\]
because $\nu\in H_{m+1}$ and by induction hypothesis ord$_t(z\circ\gammaamma)>[\pihirac{m}{n}]$ (and therefore $\gammaeq\pihirac{m+1}{n}$). Hence these monomials do not contribute to $F_\nu^{(m+1)}$ and the result follows by the quasi-homogeneity of $F_\nu^{(m+1)}$.\pisifill$\Box$
For $\nu\in L_m$ the geometry of $C_m^\nu$ is much more complicated, the ideal $J_m^\nu$ is described in Corollary \rhoef{Corolario3}.
In the next Proposition we compare jet schemes of a quasi-ordinary singularity with jet schemes of its approximated roots.
For $1\lambdaeq i\lambdaeq g$, $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we denote by $D_{i,m}^\nu$ the set
\[D_{i,m}^\nu=\{\gammaamma\in X_m^{(i)}\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k, k=1,2\}_{red},\]
and we denote $D_{g,m}^\nu$ simply by $D_m^\nu$ (see
Definition \rhoef{SemiX} for the definition of $X^{(i)}$). | 3,981 | 114,405 | en |
train | 0.29.9 | For $\nu\in H_m$ the sets $C_m^\nu$ are very easy to describe, as we see in the next Proposition.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m$ we have that
\[J_m^\nu=(z^{(0)},\lambdadots,z^{([m/n])}),\]
and hence the set $C_m^\nu$ is defined by hyperplane coordinates in $\mathbb A_m^3$ as
\[C_m^\nu=V(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)},i=1,2;z^{(0)},\lambdadots,z^{([\pihirac{m}{n}])}).\]
\lambdaabel{CHm}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
The proof is by induction on $m$. For $m=1$ we have
\[H_1=\lambdaeft\{\betaegin{array}{ll}
\{(1,1)\} & \mbox{ if }\gammaamma_1=(\pihirac{1}{n},\pihirac{1}{n})\mbox{ and }g=1\\
\\
\{(1,0),(1,1)\} & \mbox{ if }g_1>0\mbox{ (recall that the branch is normalized)}\\
\\
\{(1,0),(1,1),(0,1)\} &\mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
and the claim follows, since
\[C_1^{(1,0)}=V(x_1^{(0)},z^{(0)}),\ C_1^{(1,1)}=V(x_1^{(0)},x_2^{(0)},z^{(0)})\mbox{ and }\ C_1^{(0,1)}=V(x_2^{(0)},z^{(0)})\mbox{ if }g_1=0.\]
Suppose the claim is true for $m$ and we will prove it for $m+1$. Given $\nu\in H_{m+1}$, since $\nu\in H_m$, by induction hypothesis we have that
\[J_{m+1}^\nu=(z^{(0)},\lambdadots,z^{([m/n])},F_\nu^{(m+1)}).\]
We claim that
\[F_\nu^{(m+1)}=\lambdaeft\{\betaegin{array}{cl}
0 & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
{z^{(\pihirac{m+1}{n})}}^n & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
which proves the result, since
\[\lambdaeft[\pihirac{m+1}{n}\rhoight]=\lambdaeft\{\betaegin{array}{cl}
[\pihirac{m}{n}] & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
\pihirac{m+1}{n} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
If $F_\nu^{(m+1)}\neq 0$ then it is a quasi-homogeneous polynomial of degree $m+1$. By the expansion (\rhoef{grgr1}) given in Lemma \rhoef{LemExpSR}
\[f=z^n+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1>n_1e_1\gammaamma_1=n\gammaamma_1$. Then any monomial in $f-z^n$ verifies for any $\gammaamma\in D_m^\nu$
\[\betaegin{array}{ll}
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\mbox{ord}_t(z\circ\gammaamma)\\
& \gammaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\pihirac{m+1}{n}\\
& \gammaeq n\lambdaangle\nu,\gammaamma_1\rhoangle-r_1\lambdaangle\nu,\gammaamma_1\rhoangle+r_1\pihirac{m+1}{n}\\
& =(n-r_1)\pihirac{l_1(\nu)}{n}+r_1\pihirac{m+1}{n}>(n-r_1)\pihirac{m+1}{n}+r_1\pihirac{m+1}{n}=m+1\\
\varepsilonnd{array}\]
because $\nu\in H_{m+1}$ and by induction hypothesis ord$_t(z\circ\gammaamma)>[\pihirac{m}{n}]$ (and therefore $\gammaeq\pihirac{m+1}{n}$). Hence these monomials do not contribute to $F_\nu^{(m+1)}$ and the result follows by the quasi-homogeneity of $F_\nu^{(m+1)}$.\pisifill$\Box$
For $\nu\in L_m$ the geometry of $C_m^\nu$ is much more complicated, the ideal $J_m^\nu$ is described in Corollary \rhoef{Corolario3}.
In the next Proposition we compare jet schemes of a quasi-ordinary singularity with jet schemes of its approximated roots.
For $1\lambdaeq i\lambdaeq g$, $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we denote by $D_{i,m}^\nu$ the set
\[D_{i,m}^\nu=\{\gammaamma\in X_m^{(i)}\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k, k=1,2\}_{red},\]
and we denote $D_{g,m}^\nu$ simply by $D_m^\nu$ (see
Definition \rhoef{SemiX} for the definition of $X^{(i)}$).
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we have that
\[D_m^\nu=(\pii_{m,[\pihirac{m}{e_j}]}^a)^{-1}(D_{j,[\pihirac{m}{e_j}]}^\nu)\]
where $j=j(m,\nu)$, and for $q>p$, $\pii_{q,p}^a:\mathbb A^3_q\lambdaongrightarrow\mathbb A^3_p$ is the projection on the jet schemes of the affine ambient space.
\lambdaabel{Cgeom}
\varepsilonnd{Pro}
Hence, for $m\in\mathbb Z_{>0}$ and $\nu\in L_m$ with $j(m,\nu)=j$, the geometry of $C_m^\nu$ is determined
by the geometry of the $j$-th approximated root.
Before proving the Proposition we need the following technical result, whose proof is moved to Section \rhoef{Proofs}.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we have that for all $\gammaamma\in D_m^\nu$,
\[\betaegin{array}{l}
\mbox{ord}_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle,\ \mbox{ for }0\lambdaeq k\lambdaeq j(m,\nu)-1\\
\\
\mbox{ord}_t(f_k\circ\gammaamma)>\pihirac{m}{e_k},\ \mbox{ for }j(m,\nu)\lambdaeq k\lambdaeq g\\
\varepsilonnd{array}\]
\lambdaabel{Lemfk}
\varepsilonnd{Lem}
{\betaf\varepsilonm Proof of Proposition \rhoef{Cgeom}.} For $\nu\in H_m$ we have $j(m,\nu)=0$, and the claim follows by Proposition \rhoef{CHm}. For $\nu\in L_m$ it is enough to prove that, if $j(m,\nu)=j$ we have
\betaegin{equation}
D_m^\nu=\lambdaeft\{\gammaamma\in\mathbb A_m^3\ |\ \mbox{ord}_t(x_i\circ\gammaamma)=\nu_i, i=1,2\mbox{ and }\mbox{ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}\rhoight\}.
\lambdaabel{Claim2}
\varepsilonnd{equation}
By Lemma \rhoef{Lemfk} it follows that
\[D_m^\nu\sigmaubseteq\lambdaeft\{\gammaamma\in\mathbb A_m^3\ |\ \mbox{ord}_t(x_i\circ\gammaamma)=\nu_i,i=1,2\mbox{ and ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}\rhoight\}\]
We prove the other inclusion. Let $\gammaamma$ be a jet with ord$_t(x_i\circ\gammaamma)=\nu_i$ for $i=1,2$ and ord$_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}$. We want to prove that it is indeed an $m$-jet in $X$, or in other words, that ord$_t(f\circ\gammaamma)\gammaeq m+1$. Notice that if $j=g$ there is nothing to prove.
Then $j<g$, and first we will prove that
\[\mbox{ord}_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}.\]
Indeed, consider $f_j$ quasi-ordinary surface with $j$ characteristic exponents. If $\betaar m:=[\pihirac{m}{e_j}]$ and $\betaar\gammaamma=\pii_{m,\betaar m}(\gammaamma)$, then we have that $\betaar\gammaamma\in D_{j,\betaar m}^\nu$. Moreover $n_j\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq \betaar m$, and then, by Lemma \rhoef{Lemfk} applied to $f_j$, we have
\[\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle,\ \mbox{ for }0\lambdaeq k\lambdaeq j-1.\]
Since $\lambdaangle\nu,\gammaamma_{k+1}\rhoangle\lambdaeq\lambdaangle\nu,\gammaamma_j\rhoangle<\betaar m<m$, we deduce that ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)$.
Now we consider $f_{j+1}$. By Lemma \rhoef{Lema35} we have
\[f_{j+1}=f_j^{n_{j+1}}-c_{j+1}x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}\]
and using that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k\lambdaeq j-1$ we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_j^{n_{j+1}}\circ\gammaamma) & =n_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1^{(j+1)},\alphalpha_2^{(j+1)})+r_1^{(j+1)}\gammaamma_1+\cdots+r_j^{(j+1)}\gammaamma_j\rhoangle\\
& =n_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle >\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\varepsilonnd{array}\]
If ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$, then there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
\[\mbox{ord}_t(f_{j+1}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}\]
and we get the following inequalities
\[\betaegin{array}{rl}
(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j} & \lambdaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\pihirac{m}{e_j}\\
\\
& <\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\\
& \lambdaeq\pihirac{m}{e_{j+1}}\\
\varepsilonnd{array}\]
Hence $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j}<\pihirac{m}{e_{j+1}}=n_{j+1}\pihirac{m}{e_j}$, and then $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<(n_{j+1}-r_{j+1})\pihirac{m}{e_j}$. Since $r_{j+1}<n_{j+1}$, we have
\[\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<\pihirac{m}{e_j}\]
which is in contradiction with $j(m,\nu)=j$. Therefore we have just proved that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$. | 3,818 | 114,405 | en |
train | 0.29.10 | Indeed, consider $f_j$ quasi-ordinary surface with $j$ characteristic exponents. If $\betaar m:=[\pihirac{m}{e_j}]$ and $\betaar\gammaamma=\pii_{m,\betaar m}(\gammaamma)$, then we have that $\betaar\gammaamma\in D_{j,\betaar m}^\nu$. Moreover $n_j\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq \betaar m$, and then, by Lemma \rhoef{Lemfk} applied to $f_j$, we have
\[\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle,\ \mbox{ for }0\lambdaeq k\lambdaeq j-1.\]
Since $\lambdaangle\nu,\gammaamma_{k+1}\rhoangle\lambdaeq\lambdaangle\nu,\gammaamma_j\rhoangle<\betaar m<m$, we deduce that ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)$.
Now we consider $f_{j+1}$. By Lemma \rhoef{Lema35} we have
\[f_{j+1}=f_j^{n_{j+1}}-c_{j+1}x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}\]
and using that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k\lambdaeq j-1$ we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_j^{n_{j+1}}\circ\gammaamma) & =n_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1^{(j+1)},\alphalpha_2^{(j+1)})+r_1^{(j+1)}\gammaamma_1+\cdots+r_j^{(j+1)}\gammaamma_j\rhoangle\\
& =n_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle >\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\varepsilonnd{array}\]
If ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$, then there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
\[\mbox{ord}_t(f_{j+1}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}\]
and we get the following inequalities
\[\betaegin{array}{rl}
(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j} & \lambdaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\pihirac{m}{e_j}\\
\\
& <\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\\
& \lambdaeq\pihirac{m}{e_{j+1}}\\
\varepsilonnd{array}\]
Hence $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j}<\pihirac{m}{e_{j+1}}=n_{j+1}\pihirac{m}{e_j}$, and then $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<(n_{j+1}-r_{j+1})\pihirac{m}{e_j}$. Since $r_{j+1}<n_{j+1}$, we have
\[\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<\pihirac{m}{e_j}\]
which is in contradiction with $j(m,\nu)=j$. Therefore we have just proved that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$.
To finish, by Lemma \rhoef{LemExpSR} we have
\[f=f_{j+1}^{e_{j+1}}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_{j+1}e_{j+1}\gammaamma_{j+1}} c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k,\]
and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{j+1}^{e_{j+1}}\circ\gammaamma) & =e_{j+1}\mbox{ord}_t(f_{j+1}\circ\gammaamma)>m\\
\\
\mbox{ord}_t((c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k)\circ\gammaamma) & \gammaeq n_{j+1}e_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle=l_{j+1}(\nu)>m\\
\varepsilonnd{array}\]
Hence ord$_t(f\circ\gammaamma)>m$ as we wanted to prove. \pisifill$\Box$
As a consequence of Proposition \rhoef{Cgeom}, we have the following algebraic counterpart, where we explain how the equations of the approximated roots appear as generators of $J_m^\nu$, and therefore a minimal presentation of the ideal $J_m^\nu$ is given.
\betaegin{Cor} Given $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, for $l_i(\nu)\lambdaeq l<l_{i+1}(\nu)$ (resp. $l_i(\nu)\lambdaeq l\lambdaeq m$) if $c(\nu)\lambdaeq i< j(m,\nu)$ (resp. $i=j(m,\nu)$), we have that
\[F_\nu^{(l)}=\lambdaeft\{\betaegin{array}{cl}
{F_{i,\nu}^{(\pihirac{l}{e_i})}}^{e_i} & \mbox{ if }l\varepsilonquiv 0\mbox{ mod }e_i\\
\\
0 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
Hence the ideal $J_m^\nu$ is generated by the polynomials
\[J_m^\nu=\lambdaeft(F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+k_i)}\rhoight)_i\]
for $c(\nu)\lambdaeq i\lambdaeq j(m,\nu)$ such that $l_i(\nu)<l_{i+1}(\nu)$, and $0\lambdaeq k_i<\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}\mbox{ if }i<j(m,\nu)$ and $0\lambdaeq k_{j(m,\nu)}\lambdaeq
[\pihirac{m-l_{j(m,\nu)}(\nu)}{e_{j(m,\nu)}}]$. Moreover, we have
\betaegin{equation}
F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+G_{i,\nu},
\lambdaabel{EqP}
\varepsilonnd{equation}
where $G_{i,\nu}$ is the polynomial
\[G_{i,\nu}=\sigmaum c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}{F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{r_i}\]
with $c_{\underline{\alphalpha},\underline{r}}$ are the coefficients appearing in the expansion given in Lemma \rhoef{Lema35}, and such that
\betaegin{enumerate}
\item[(i)] $c_{\underline{\alphalpha},\underline{r}}\neq 0$
\item[(ii)] $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$
\varepsilonnd{enumerate}
Notice that condition (ii) does not hold when $\nu\notin\rho_1\cup\rho_2$, and hence $G_{i,\nu}=0$ in these cases.
\lambdaabel{Corolario3}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It is a consequence of Proposition \rhoef{Cgeom}, applied to any $l_i(\nu)\lambdaeq l<l_{i+1}(\nu)$, and using the trivial observation that $J_{m'}^\nu\sigmaubseteq J_m^\nu$ for any $m'<m$. Indeed, for any $l_i(\nu)\lambdaeq l< l_{i+1}(\nu)$ we study the polynomials $F_\nu^{(l_i(\nu))},\lambdadots,F_\nu^{(l)}$ (note that we need $l_i(\nu)<l_{i+1}(\nu)$). We have that $j(l,\nu)=i$, and, by Proposition \rhoef{Cgeom}, $D_l^\nu=(\pii_{l,[\pihirac{l}{e_i}]}^a)^{-1}(D_{i,[\pihirac{l}{e_i}]}^\nu)$, or in other words
\[D_l^\nu=\lambdaeft\{\gammaamma\in\mathbb A_l^3\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k,\ k=1,2,\mbox{ ord}_t(f_i\circ\gammaamma)>\lambdaeft[\pihirac{l}{e_i}\rhoight]\rhoight\}\]
Then the ideal $J_l^\nu$ only depends on $f_i$ (and hence on its approximated roots). Moreover, by Lemma \rhoef{Lemfk}, we deduce that for $0\lambdaeq k<i$
\[F_{k,\nu}^{(r_k)}\in J_l^\nu,\ \mbox{ for }0\lambdaeq r_k<\lambdaangle\nu,\gammaamma_{k+1}\rhoangle.\]
By the expansion (\rhoef{grgr1}) in Lemma \rhoef{LemExpSR}
\[f=f_i^{e_i}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$.
The part $f_i^{e_i}$ contributes to $F^{(l)}$ with ${F_{i,\nu}^{(\pihirac{l}{e_i})}}^{e_i}$ and only when $l$ is divisible by $e_i$. While for the {\varepsilonm monomials}
$x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}$ the contribution is given by
\[{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots {F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{r_i}{F_{i,\nu}^{(\pihirac{a}{r_{i+1}})}}^{r_{i+1}}\]
with $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+a=l$, and only when $a$ is divisible by $r_{i+1}$. Set $a=br_{i+1}$, we claim that $F_{i,\nu}^{(b)}$ belongs to $J_l^\nu$ and hence the {\varepsilonm monomial} does not contribute to $F^{(l)}$. Indeed, at level $l'=be_i$ it appears as ${F_{i,\nu}^{(b)}}^{e_i}$, and we only need to prove that $l'<l$. Since
\[\betaegin{array}{ll}
l & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}b\\
& =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}\pihirac{l'}{e_i}\\
\varepsilonnd{array}\]
we have
\[\betaegin{array}{ll}
l' & =\pihirac{e_i}{r_{i+1}}(l-\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle)\\
& \lambdaeq\pihirac{e_i}{r_{i+1}}(l-l_{i+1}(\nu)+r_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle)\\
& =\pihirac{e_i}{r_{i+1}}l-(\pihirac{e_i}{r_{i+1}}-1)l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Therefore $r_{i+1}l'\lambdaeq e_il-(e_i-r_{i+1})l_{i+1}(\nu)$. Suppose that $l'\gammaeq l$, then
\[(e_i-r_{i+1})l_{i+1}(\nu)\lambdaeq e_il-r_{i+1}l'\lambdaeq (e_i-r_{i+1})l\]
and since $e_i>r_{i+1}$ it contradicts the fact that $j(l,\nu)=i$. | 4,070 | 114,405 | en |
train | 0.29.11 | \betaegin{enumerate}
\item[(i)] $c_{\underline{\alphalpha},\underline{r}}\neq 0$
\item[(ii)] $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$
\varepsilonnd{enumerate}
Notice that condition (ii) does not hold when $\nu\notin\rho_1\cup\rho_2$, and hence $G_{i,\nu}=0$ in these cases.
\lambdaabel{Corolario3}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It is a consequence of Proposition \rhoef{Cgeom}, applied to any $l_i(\nu)\lambdaeq l<l_{i+1}(\nu)$, and using the trivial observation that $J_{m'}^\nu\sigmaubseteq J_m^\nu$ for any $m'<m$. Indeed, for any $l_i(\nu)\lambdaeq l< l_{i+1}(\nu)$ we study the polynomials $F_\nu^{(l_i(\nu))},\lambdadots,F_\nu^{(l)}$ (note that we need $l_i(\nu)<l_{i+1}(\nu)$). We have that $j(l,\nu)=i$, and, by Proposition \rhoef{Cgeom}, $D_l^\nu=(\pii_{l,[\pihirac{l}{e_i}]}^a)^{-1}(D_{i,[\pihirac{l}{e_i}]}^\nu)$, or in other words
\[D_l^\nu=\lambdaeft\{\gammaamma\in\mathbb A_l^3\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k,\ k=1,2,\mbox{ ord}_t(f_i\circ\gammaamma)>\lambdaeft[\pihirac{l}{e_i}\rhoight]\rhoight\}\]
Then the ideal $J_l^\nu$ only depends on $f_i$ (and hence on its approximated roots). Moreover, by Lemma \rhoef{Lemfk}, we deduce that for $0\lambdaeq k<i$
\[F_{k,\nu}^{(r_k)}\in J_l^\nu,\ \mbox{ for }0\lambdaeq r_k<\lambdaangle\nu,\gammaamma_{k+1}\rhoangle.\]
By the expansion (\rhoef{grgr1}) in Lemma \rhoef{LemExpSR}
\[f=f_i^{e_i}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$.
The part $f_i^{e_i}$ contributes to $F^{(l)}$ with ${F_{i,\nu}^{(\pihirac{l}{e_i})}}^{e_i}$ and only when $l$ is divisible by $e_i$. While for the {\varepsilonm monomials}
$x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}$ the contribution is given by
\[{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots {F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{r_i}{F_{i,\nu}^{(\pihirac{a}{r_{i+1}})}}^{r_{i+1}}\]
with $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+a=l$, and only when $a$ is divisible by $r_{i+1}$. Set $a=br_{i+1}$, we claim that $F_{i,\nu}^{(b)}$ belongs to $J_l^\nu$ and hence the {\varepsilonm monomial} does not contribute to $F^{(l)}$. Indeed, at level $l'=be_i$ it appears as ${F_{i,\nu}^{(b)}}^{e_i}$, and we only need to prove that $l'<l$. Since
\[\betaegin{array}{ll}
l & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}b\\
& =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}\pihirac{l'}{e_i}\\
\varepsilonnd{array}\]
we have
\[\betaegin{array}{ll}
l' & =\pihirac{e_i}{r_{i+1}}(l-\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle)\\
& \lambdaeq\pihirac{e_i}{r_{i+1}}(l-l_{i+1}(\nu)+r_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle)\\
& =\pihirac{e_i}{r_{i+1}}l-(\pihirac{e_i}{r_{i+1}}-1)l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Therefore $r_{i+1}l'\lambdaeq e_il-(e_i-r_{i+1})l_{i+1}(\nu)$. Suppose that $l'\gammaeq l$, then
\[(e_i-r_{i+1})l_{i+1}(\nu)\lambdaeq e_il-r_{i+1}l'\lambdaeq (e_i-r_{i+1})l\]
and since $e_i>r_{i+1}$ it contradicts the fact that $j(l,\nu)=i$.
Now equation (\rhoef{EqP}) follows by Lemma \rhoef{Lema35}.
\pisifill$\Box$
From Corollary \rhoef{Corolario3} we deduce the following:
\betaegin{equation}
D_m^\nu\sigmaubset D(F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}\cdots F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}),\ \mbox{ for any }\nu\in L_m\mbox{ with }j(m,\nu)=j.
\lambdaabel{eqO}
\varepsilonnd{equation}
To illustrate the description of $J_m^\nu$ given in Corollary \rhoef{Corolario3}, we consider some particular cases.
$\betaullet$ First, the simplest case, when $\nu\notin\rho_1\cup\rho_2$ we have $G_{j,\nu}=0$ for any $j$, and $m(\nu)=1$. Hence (note that we also have $c(\nu)=0$):
\betaegin{equation}
\betaegin{array}{lcl}
F_{0,\nu}^{(r)} & = & z^{(r)}, \mbox{ for }0\lambdaeq r<\pihirac{l_{1}(\nu)}{e_0}\\
\\
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})} & = & {z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}\\
\\
F_1^{(\pihirac{l_1(\nu)}{e_1}+r)} & &\mbox{ for }1\lambdaeq r<\pihirac{l_2(\nu)-l_1(\nu)}{e_1}\\
\ \ \ \ \ \vdots & \vdots & \\
F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g-1}})} & = & {F_{g_1-2,\nu}}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-2}})}-{x_1^{(\nu_1)}}^{n_{g_1-1}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1-1)}}\cdots
{F_{g_1-3,\nu}^{(\pihirac{l_{g_1-2}(\nu)}{e_{g_1-3}})}}^{r_{g_1-2}^{(g_1-1)}}\\
\\
F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g-1}}+r)} & & \mbox{ for }1\lambdaeq r<\pihirac{l_{g_1}(\nu)-l_{g_1-1}(\nu)}{e_{g_1-1}}\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}})} & = & {F_{g_1-1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{n_{g_1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1)}}{z^{(\lambdaangle\nu,
\gammaamma_1\rhoangle)}}^{r_1^{(g_1)}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}^{r_2^{(g_1)}} \cdots {F_{g_1-2,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-2}})}}^{r_{g_1-1}^{(g_1)}}\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+r)} & &\mbox{ for }1\lambdaeq r<\pihirac{l_{g_1+1}(\nu)-l_{g_1}(\nu)}{e_{g_1}}\\
\\
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & = & {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}
{x_2^{(\nu_2)}}^{\alphalpha_2^{(g_1+1)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}\\
\ \ \ \ \ \vdots & \vdots &\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j})} & = & {F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}}^{n_j}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(j)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(j)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(j)}}\cdots {F_{j-2,\nu}^{(\pihirac{l_{j-1}(\nu)}{e_{j-2}})}}^{r_{j-1}^{(j)}}\\
\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j}+r)} & & \mbox{ for }1\lambdaeq r\lambdaeq[\pihirac{m-l_j(\nu)}{e_j}]\\
\varepsilonnd{array}
\lambdaabel{ecsFs}
\varepsilonnd{equation}
Notice that the variable $x_2^{(\nu_2)}$ appears for the first time in the equation $F_{g_1+1}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}$, and raised to the power one or bigger depending on whether $g_2=g_1+1$ or $g_2=g_1$ respectively.
$\betaullet$ Another example is when $j(m,\nu)=c(\nu)$, then
\[J_m^\nu=\lambdaeft(F_{c(\nu),\nu}^{(0)},\lambdadots,F_{c(\nu),\nu}^{([\pihirac{m}{e_{c(\nu)}}])}\rhoight)\]
and if moreover $c(\nu)=0$ this is just the content of Proposition \rhoef{CHm}. Notice that the description of $J_m^\nu$ for $\nu\in H_m$ given in Proposition \rhoef{CHm} is contained in Corollary \rhoef{Corolario3}, but we wanted to stress the fact that for $\nu\in H_m$ the description is particularly simple.
Now we can prove the irreducibility of the sets $C_m^\nu$.
\betaegin{Pro}
For any $m\in\mathbb Z_{> 0}$ and $\nu\in H_m\cup L_m$, the set $C_m^\nu$ is irreducible and has codimension
\[\nu_1+\nu_2+\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}+\lambdaeft[\pihirac{m-l_{j(m,\nu)}(\nu)}{e_{j(m,\nu)}}\rhoight]+1\]
\lambdaabel{Prop1}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
To simplify notation we will denote along this proof $j(m,\nu)$ just by $j$ and by $k_i(\nu)$, or simply by $k_i$ when $\nu$ is clear from the context, we denote
the quotient $\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}$.
First notice that, by definition of $c(\nu)$, $\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}=\sigmaum_{k=c(\nu)}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}$.
If $\nu\in H_m$ we have $j(m,\nu)=0$ and the claim about the codimension follows by Proposition \rhoef{CHm}. The closed set $C_m^\nu$ is
irreducible since it is defined by hyperplane coordinates.
If $\nu\in L_m^=$, then by Lemma \rhoef{LemCm}, we have that $C_m^\nu=V(I^\nu,J_m^\nu)$, where the ideal $J_m^\nu$ is described in equation (\rhoef{eqJ}). Then
Codim$(C_m^\nu)=\nu_1+\nu_2+\pihirac{l_1(\nu)}{n}+1$. Notice that if $c(\nu)>0$ then $l_1(\nu)=0$. The
irreducibility of $C_m^\nu$ follows from the irreducibility of $F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}$. | 3,832 | 114,405 | en |
train | 0.29.12 | \ \ \ \ \ \vdots & \vdots &\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j})} & = & {F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}}^{n_j}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(j)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(j)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(j)}}\cdots {F_{j-2,\nu}^{(\pihirac{l_{j-1}(\nu)}{e_{j-2}})}}^{r_{j-1}^{(j)}}\\
\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j}+r)} & & \mbox{ for }1\lambdaeq r\lambdaeq[\pihirac{m-l_j(\nu)}{e_j}]\\
\varepsilonnd{array}
\lambdaabel{ecsFs}
\varepsilonnd{equation}
Notice that the variable $x_2^{(\nu_2)}$ appears for the first time in the equation $F_{g_1+1}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}$, and raised to the power one or bigger depending on whether $g_2=g_1+1$ or $g_2=g_1$ respectively.
$\betaullet$ Another example is when $j(m,\nu)=c(\nu)$, then
\[J_m^\nu=\lambdaeft(F_{c(\nu),\nu}^{(0)},\lambdadots,F_{c(\nu),\nu}^{([\pihirac{m}{e_{c(\nu)}}])}\rhoight)\]
and if moreover $c(\nu)=0$ this is just the content of Proposition \rhoef{CHm}. Notice that the description of $J_m^\nu$ for $\nu\in H_m$ given in Proposition \rhoef{CHm} is contained in Corollary \rhoef{Corolario3}, but we wanted to stress the fact that for $\nu\in H_m$ the description is particularly simple.
Now we can prove the irreducibility of the sets $C_m^\nu$.
\betaegin{Pro}
For any $m\in\mathbb Z_{> 0}$ and $\nu\in H_m\cup L_m$, the set $C_m^\nu$ is irreducible and has codimension
\[\nu_1+\nu_2+\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}+\lambdaeft[\pihirac{m-l_{j(m,\nu)}(\nu)}{e_{j(m,\nu)}}\rhoight]+1\]
\lambdaabel{Prop1}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
To simplify notation we will denote along this proof $j(m,\nu)$ just by $j$ and by $k_i(\nu)$, or simply by $k_i$ when $\nu$ is clear from the context, we denote
the quotient $\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}$.
First notice that, by definition of $c(\nu)$, $\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}=\sigmaum_{k=c(\nu)}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}$.
If $\nu\in H_m$ we have $j(m,\nu)=0$ and the claim about the codimension follows by Proposition \rhoef{CHm}. The closed set $C_m^\nu$ is
irreducible since it is defined by hyperplane coordinates.
If $\nu\in L_m^=$, then by Lemma \rhoef{LemCm}, we have that $C_m^\nu=V(I^\nu,J_m^\nu)$, where the ideal $J_m^\nu$ is described in equation (\rhoef{eqJ}). Then
Codim$(C_m^\nu)=\nu_1+\nu_2+\pihirac{l_1(\nu)}{n}+1$. Notice that if $c(\nu)>0$ then $l_1(\nu)=0$. The
irreducibility of $C_m^\nu$ follows from the irreducibility of $F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}$.
Now let $\nu$ be an element in $L_m^<$. We have to study carefully the generators of $J_m^\nu$ given in Corollary \rhoef{Corolario3}. Any $F_{i,\nu}^{(l)}$ is quasi-homogeneous of degree $l$, but the second property described in Remark \rhoef{RemTonto} is not true anymore once we consider the equations modulo $I^\nu$. We need to know when a certain variable $x_k^{(l)}$ or $z^{(l)}$ appear for the first time in the generators of $J_m^\nu$. Notice that for any $\gammaamma\in D_m^\nu$ we have ord$_t(x_i\circ\gammaamma)=\nu_i$ for $i=1,2$, and ord$_t(z\circ\gammaamma)=\lambdaangle\nu,\gammaamma_1\rhoangle$. It is clear that the variables $x_k^{(\nu_k)}$ for $k=1,2$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$ appear for the first time in the first non-monomial equation $F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}$. In the next equation appear $x_k^{(\nu_k+1)}$, $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}$, and so on. Then looking at the generators of $J_m^\nu$ described in Corollary \rhoef{Corolario3}, we deduce that, for $m(\nu)\lambdaeq i<j$ and $0\lambdaeq r<k_i$, or $0\lambdaeq r<[\pihirac{m-l_j(\nu)}{e_j}]$ when $i=j$,
\[(\alphast)\ \ \ \ \ \ \ \mbox{ the variables }x_k^{(\nu_k+k_{m(\nu)}+\cdots+k_{i-1}+r)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k_{m(\nu)}+\cdots+k_{i-1}+r)}\mbox{ appear for the first time in }F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\]
Notice that for $1\lambdaeq l<m(\nu)$ we have $k_l=0$.
We divide the set of generators of $J_m^\nu$, given in Corollary \rhoef{Corolario3}, in two sets:
\[\betaegin{array}{l}
\mathcal C_1=\lambdaeft\{F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}\rhoight\}_{c(\nu)\lambdaeq i\lambdaeq j,\ l_i(\nu)<l_{i+1}(\nu)}\\
\\
\mathcal C_2=\lambdaeft\{F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\rhoight\}_{(i,r)\in A_2}\\
\varepsilonnd{array}\]
where $A_2=\{(i,r)\ |\ c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu),\ 0<r<k_i\}\cup\{(j,r)\ |\ 0<r<[\pihirac{m-l_j(\nu)}{e_j}]\}$.
We claim:
\betaegin{enumerate}
\item[(i)] $V(\mathcal C_1)\sigmaimeq Z^{\Gamma_m^\nu}$, the toric variety defined by the semigroup $\Gamma_m^\nu$ generated by
\[\{\gammaamma_i\}_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\]
If $\nu\notin\rho_1\cup\rho_2$ then $\Gamma_m^\nu=\Gamma_{j(m,\nu)}$ and $V(\mathcal C_1)$ is isomorphic to the monomial variety associated to $X^{(j(m,\nu))}$ (see Definition \rhoef{defMV}).
\
\item[(ii)] any $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\in\mathcal C_2$ is linear over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$ with respect to one of the variables described in ($\alphast$), which appears for the first time on this equation.
\varepsilonnd{enumerate}
Since any of these equations in $\mathcal C_2$ is linear in a different variable, and, by ($\alphast$) we have that it appears for the first time in $\mathcal C_2$, we deduce
\[V(I^\nu,\mathcal C_2)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaimeq\mathbb A^{\alphalpha(m,\nu)}\]
where $\alphalpha(m,\nu)=3(m+1)-\nu_1-\nu_2-\sigmaum_{i=c(\nu),...,j-1,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)-[\pihirac{m-l_j(\nu)}{e_j}]$, because $V(I^\nu,\mathcal C_2)\sigmaubseteq\mathbb A_m^3\sigmaimeq\mathbb A^{3(m+1)}$. Hence
\[D_m^\nu\sigmaimeq\lambdaeft(Z^{\Gamma_m^\nu}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\rhoight)\tauimes\mathbb A^{\alphalpha(m,\nu)}\]
The toric variety $Z^{\Gamma_m^\nu}$ is irreducible and hence the irreducibility of $C_m^\nu$ follows by the previous isomorphism. Moreover $Z^{\Gamma_m^\nu}$ is complete intersection, hence the codimension equals the number of defining equations, which is the cardinal of $\mathcal C_1$.
Therefore
\[\betaegin{array}{ll}
\mbox{Codim}(C_m^\nu) & =\sigmaharp\mathcal C_1+\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)+[\pihirac{m-l_j(\nu)}{e_j}]\\
\\
& =\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i+[\pihirac{m-l_j(\nu)}{e_j}]+\sigmaharp\mathcal C_1-\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}\\
\varepsilonnd{array}\]
Finally the statement about the codimension follows now by these two remarks:
$\betaullet$ $\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i=\sigmaum_{i=c(\nu)}^{j-1}k_i$, since $k_i=0$ whenever $l_i(\nu)=l_{i+1}(\nu)$.
$\betaullet$ $l_j(\nu)<l_{j+1}(\nu)$ by definition of $j(m,\nu)$, and therefore
\[\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}=\sigmaharp\mathcal C_1-1.\]
Now we prove the claim. To prove (i), notice that we can write equation (\rhoef{EqP}) as
\[F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\cdot U\]
where, arguing as in the proof of Lemma \rhoef{TechLem} (i), we have $U\neq 0$. Then we have the isomorphism
\[V(F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\sigmaimeq V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\]
where $h_i=w_i^{n_i}-x_1^{\alphalpha_1^{(i)}}
x_2^{\alphalpha_2^{(i)}}z^{r_1^{(i)}}w_1^{r_2^{(i)}}\cdots w_{i-2}^{r_{i-1}^{(i)}}$, with the relation $n_i\gammaamma_i=(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1^{(i)}\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}$. And $V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}$ is isomorphic to the toric variety $Z^{\Gamma_m^\nu}$.
To prove the claim (ii), we distinguish three cases, depending on $m$. | 3,452 | 114,405 | en |
train | 0.29.13 | \
\item[(ii)] any $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\in\mathcal C_2$ is linear over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$ with respect to one of the variables described in ($\alphast$), which appears for the first time on this equation.
\varepsilonnd{enumerate}
Since any of these equations in $\mathcal C_2$ is linear in a different variable, and, by ($\alphast$) we have that it appears for the first time in $\mathcal C_2$, we deduce
\[V(I^\nu,\mathcal C_2)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaimeq\mathbb A^{\alphalpha(m,\nu)}\]
where $\alphalpha(m,\nu)=3(m+1)-\nu_1-\nu_2-\sigmaum_{i=c(\nu),...,j-1,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)-[\pihirac{m-l_j(\nu)}{e_j}]$, because $V(I^\nu,\mathcal C_2)\sigmaubseteq\mathbb A_m^3\sigmaimeq\mathbb A^{3(m+1)}$. Hence
\[D_m^\nu\sigmaimeq\lambdaeft(Z^{\Gamma_m^\nu}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\rhoight)\tauimes\mathbb A^{\alphalpha(m,\nu)}\]
The toric variety $Z^{\Gamma_m^\nu}$ is irreducible and hence the irreducibility of $C_m^\nu$ follows by the previous isomorphism. Moreover $Z^{\Gamma_m^\nu}$ is complete intersection, hence the codimension equals the number of defining equations, which is the cardinal of $\mathcal C_1$.
Therefore
\[\betaegin{array}{ll}
\mbox{Codim}(C_m^\nu) & =\sigmaharp\mathcal C_1+\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)+[\pihirac{m-l_j(\nu)}{e_j}]\\
\\
& =\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i+[\pihirac{m-l_j(\nu)}{e_j}]+\sigmaharp\mathcal C_1-\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}\\
\varepsilonnd{array}\]
Finally the statement about the codimension follows now by these two remarks:
$\betaullet$ $\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i=\sigmaum_{i=c(\nu)}^{j-1}k_i$, since $k_i=0$ whenever $l_i(\nu)=l_{i+1}(\nu)$.
$\betaullet$ $l_j(\nu)<l_{j+1}(\nu)$ by definition of $j(m,\nu)$, and therefore
\[\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}=\sigmaharp\mathcal C_1-1.\]
Now we prove the claim. To prove (i), notice that we can write equation (\rhoef{EqP}) as
\[F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\cdot U\]
where, arguing as in the proof of Lemma \rhoef{TechLem} (i), we have $U\neq 0$. Then we have the isomorphism
\[V(F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\sigmaimeq V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\]
where $h_i=w_i^{n_i}-x_1^{\alphalpha_1^{(i)}}
x_2^{\alphalpha_2^{(i)}}z^{r_1^{(i)}}w_1^{r_2^{(i)}}\cdots w_{i-2}^{r_{i-1}^{(i)}}$, with the relation $n_i\gammaamma_i=(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1^{(i)}\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}$. And $V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}$ is isomorphic to the toric variety $Z^{\Gamma_m^\nu}$.
To prove the claim (ii), we distinguish three cases, depending on $m$.
(a) For $m<l_{g_1+1}(\nu)+e_{g_1+1}$. In this case we have that $g_1>0$ and therefore $m(\nu)$ is either $g_1$ or $1$.
Suppose first that $m(\nu)=1$. Then
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})}= {z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}+G_{1,\nu},\]
where $G_{1,\nu}$ is the polynomial $\sigmaum c_{i_1i_2k}{x_1^{(\nu_1)}}^{i_1}{x_2^{(\nu_2)}}^{i_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k$ with $\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle=n_1\lambdaangle\nu,\gammaamma_1\rhoangle=a_1\nu_1$. Moreover, for $0<r<k_1$, $F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}$ is a quasi-homogeneous polynomial of degree $\pihirac{l_1(\nu)}{e_1}+r=a_1\nu_1+r$, and using ($\alphast$),
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}={x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+r)}\cdot U_1^{(r)}+H_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}\]
where $H_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}$ is a polynomial in which $x_1^{(\nu_1+r)}$ do not appear, and $U_1^{(r)}\neq 0$. Analogously, for $1<i<j(m,\nu)$ and $0<r<k_i(\nu)$ we have
\[F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}= n_i{F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i-1}F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}}+r)}+H_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)},\]
where $H_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}$ is a polynomial in which the variable $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ does not appear. Moreover
\[F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}}+r)}= a_1{x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}+H_{i-1,\nu}^{(r)},\]
where the variable $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ does not appear in the polynomial $H_{i-1,\nu}^{(r)}$. Then,
by Lemma \rhoef{TechLem} (ii) it follows that for $1=m(\nu)\lambdaeq i<j(m,\nu)$ and $0<r<k_i(\nu)$ the equation
$F_i^{(\pihirac{l_i(\nu)}{e_i}+r)}$ is linear on $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ over $D(x_1^{(\nu_1)})$.
We still have to deal with the equations $F_j^{(\pihirac{l_j(\nu)}{e_j}+r)}$ with $1\lambdaeq r\lambdaeq [\pihirac{m-l_j(\nu)}{e_j}]$. Notice that $j(m,\nu)\lambdaeq g_1+1$. If $j(m,
\nu)<g_1+1$ then the argument is exactly as before, and if $j(m,\nu)=g_1+1$ then $\lambdaeft[\pihirac{m-l_j(\nu)}{e_j}\rhoight]=0$.
Suppose now that $m(\nu)=g_1>1$. Then $\nu=(0,\nu_2)$ and the generators of $J_m^\nu$ are
\[\betaegin{array}{ll}
F_{g_1}^{(r)} \mbox{ for }0\lambdae r\lambdaeq[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}] & \mbox{ if }j(m,\nu)=g_1\\
\\
F_{g_1}^{(r)}\mbox{ for }0\lambdaeq r<k_{g_1}(\nu)\mbox{ and }F_{g_1+1}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & \mbox{ if }j(m,\nu)=g_1+1\\
\varepsilonnd{array}\]
For $r>0$ we have that $F_{g_1}^{(r)}\varepsilonquiv \pihirac{\piartial F_{g_1}}{\piartial x_1}(x_1^{(0)},z^{(0)})x_1^{(r)}+\pihirac{\piartial F_{g_1}}{\piartial z}(x_1^{(0)},
z^{(0)})z^{(0)}+H_r\mbox{ mod }I^\nu$, with $H_r$ a polynomial where the variables $x_1^{(r)}$ and $z^{(r)}$ do not appear. Since we are looking at $\{x_1^{(0)}
\neq 0\}$, we are outside the singular locus, and we deduce that these equations are linear either in $x_1^{(r)}$ or in $z^{(r)}$. The rest of the proof follows as in the previous part of this case.
(b) For $l_{g_1+1}(\nu)+e_{g_1+1}\lambdaeq m<l_{g_2+1}(\nu)+e_{g_2+1}$. This is only possible when $g_2=g_1+1$.
Notice that if $j(m,\nu)=g_2+1$ then $\lambdaeft[\pihirac{m-l_j(\nu)}{e_j}\rhoight]=0$. We just have to study the generators $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}$ for $i>g_1$
(which are all the generators of $J_m^\nu$ when $g_1=0$) since the others were studied in the previous case. That is, we have the equations $F_{g_2,\nu}^{(\pihirac{l_{g_2}
(\nu)}{e_{g_2}}+r)}$ for $0\lambdaeq r\lambdaeq k_{g_2}(\nu)-1$ if $j(m,\nu)=g_2+1$ and $0\lambdaeq r\lambdaeq\lambdaeft[\pihirac{m-l_{g_2}(\nu)}{e_{g_2}}\rhoight]$ otherwise. If $j(m,\nu)=g_2+1$
we also have the generator $F_{g_2+1,\nu}^{(\pihirac{l_{g_2+1}(\nu)}{e_{g_2+1}})}$.
\[\betaegin{array}{lcl}
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}})} & = & {F_{g_1}^{(\pihirac{l_{g_2}(\nu)}{e_{g_1}})}}^{n_{g_2}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2)}{z^{(\lambdaangle
\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots {F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}+G_{g_2,\nu}\\
\\
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)} & = & {x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2+r)}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots
{F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}\cdot U_{g_2}^{(r)}+H_r\\
\varepsilonnd{array}\]
where $U_{g_2}^{(r)}$ is $1$ if $\nu\notin\rho_1\cup\rho_2$ and $\neq 0$ otherwise, and $H_r$ is a polynomial in which the variable $x_2^{(\nu_2+r)}$ does not appear. Then by Lemma \rhoef{TechLem} (ii) we deduce that every $F_{g_2,\nu}^
{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)}$ is linear on $x_2^{(\nu_2+r)}$ over $D(x_1^{(\nu_1)})$. And the rest of the argument goes as in the previous case.
(c) For $l_{g_2+1}(\nu)+e_{g_2+1}\lambdaeq m$, with the same arguments it is easy to see that for $i>g_2$ and $1\lambdaeq r<k_i(\nu)$ each $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+
r)}$ is linear on $x_1^{(\nu_1+k_1+\cdots+k_i+r)}$ over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$.
\pisifill$\Box$
In particular we have the following variation of the codimension of $C_m^\nu$ as $m$ grows.
\betaegin{Cor}
For $\nu\in H_m\cup L_m$ such that $\nu\in H_{m-1}\cup L_{m-1}$ we have that
\[\mbox{Codim}(C_m^\nu)=\lambdaeft\{\betaegin{array}{ll}
\mbox{Codim}(C_{m-1}^\nu)+1 & \mbox{ if }m\varepsilonquiv 0\mbox{ mod }e_{j(m-1,\nu)}\\
\\
\mbox{Codim}(C_{m-1}^\nu) & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
\lambdaabel{CorCodim}
\varepsilonnd{Cor} | 4,069 | 114,405 | en |
train | 0.29.14 | (b) For $l_{g_1+1}(\nu)+e_{g_1+1}\lambdaeq m<l_{g_2+1}(\nu)+e_{g_2+1}$. This is only possible when $g_2=g_1+1$.
Notice that if $j(m,\nu)=g_2+1$ then $\lambdaeft[\pihirac{m-l_j(\nu)}{e_j}\rhoight]=0$. We just have to study the generators $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}$ for $i>g_1$
(which are all the generators of $J_m^\nu$ when $g_1=0$) since the others were studied in the previous case. That is, we have the equations $F_{g_2,\nu}^{(\pihirac{l_{g_2}
(\nu)}{e_{g_2}}+r)}$ for $0\lambdaeq r\lambdaeq k_{g_2}(\nu)-1$ if $j(m,\nu)=g_2+1$ and $0\lambdaeq r\lambdaeq\lambdaeft[\pihirac{m-l_{g_2}(\nu)}{e_{g_2}}\rhoight]$ otherwise. If $j(m,\nu)=g_2+1$
we also have the generator $F_{g_2+1,\nu}^{(\pihirac{l_{g_2+1}(\nu)}{e_{g_2+1}})}$.
\[\betaegin{array}{lcl}
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}})} & = & {F_{g_1}^{(\pihirac{l_{g_2}(\nu)}{e_{g_1}})}}^{n_{g_2}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2)}{z^{(\lambdaangle
\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots {F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}+G_{g_2,\nu}\\
\\
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)} & = & {x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2+r)}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots
{F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}\cdot U_{g_2}^{(r)}+H_r\\
\varepsilonnd{array}\]
where $U_{g_2}^{(r)}$ is $1$ if $\nu\notin\rho_1\cup\rho_2$ and $\neq 0$ otherwise, and $H_r$ is a polynomial in which the variable $x_2^{(\nu_2+r)}$ does not appear. Then by Lemma \rhoef{TechLem} (ii) we deduce that every $F_{g_2,\nu}^
{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)}$ is linear on $x_2^{(\nu_2+r)}$ over $D(x_1^{(\nu_1)})$. And the rest of the argument goes as in the previous case.
(c) For $l_{g_2+1}(\nu)+e_{g_2+1}\lambdaeq m$, with the same arguments it is easy to see that for $i>g_2$ and $1\lambdaeq r<k_i(\nu)$ each $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+
r)}$ is linear on $x_1^{(\nu_1+k_1+\cdots+k_i+r)}$ over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$.
\pisifill$\Box$
In particular we have the following variation of the codimension of $C_m^\nu$ as $m$ grows.
\betaegin{Cor}
For $\nu\in H_m\cup L_m$ such that $\nu\in H_{m-1}\cup L_{m-1}$ we have that
\[\mbox{Codim}(C_m^\nu)=\lambdaeft\{\betaegin{array}{ll}
\mbox{Codim}(C_{m-1}^\nu)+1 & \mbox{ if }m\varepsilonquiv 0\mbox{ mod }e_{j(m-1,\nu)}\\
\\
\mbox{Codim}(C_{m-1}^\nu) & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
\lambdaabel{CorCodim}
\varepsilonnd{Cor}
\sigmaubsection{Inclusions among the $C_m^\nu$}
We have a collection of irreducible sets $\{C_m^\nu\ |\ \nu\in H_m\cup L_m\}$ covering $\pii_m^{-1}(X_{Sing})$, but in general it is not its decomposition in irreducible components. We have to study the inclusions
\betaegin{equation}
C_m^{\nu'}\sigmaubseteq C_m^\nu\mbox{ for different }\nu,\nu'\in H_m\cup L_m.
\lambdaabel{ContGen}
\varepsilonnd{equation}
We need to see $C_m^\nu$ as the closure of a set, which is slightly different from $D_m^\nu$, though described by the ideals $I^\nu$ and $J_m^\nu$. For instance, when $j(m,\nu)=c(\nu)=g_1>0$, by Corollary \rhoef{Corolario3} $J_m^\nu=(F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])})$, and
\[C_m^\nu=\mathbb overline{V(I^\nu,F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])})\cap D(x_1^{(\nu_1)})},\]
because the polynomials $F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])}$ do not depend on $x_2^{(\nu_2)}$, and hence, when taking the Zariski closure, we can drop the condition $D(x_2^{(\nu_2)})$ in the description of $D_m^\nu$ given in (\rhoef{eqast}). This is the description we are looking for, and it is the content of the next Lemma.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ we have that $C_m^\nu=\mathbb overline{O_m^\nu}$, where
\[O_m^\nu:=\lambdaeft\{\betaegin{array}{cl}
V(I^\nu,J_m^\nu) & \mbox{ if }j'(m,\nu)<m(\nu)\\
\\
V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)}) & \mbox{ if }m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_2\\
\\
V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)}) & \mbox{ if }j'(m,\nu)\gammaeq g_2+1\\
\varepsilonnd{array}\rhoight.\]
Notice that when $j'(m,\nu)<m(\nu)$ then $O_m^\nu=C_m^\nu$, and if $j'(m,\nu)\gammaeq g_2+1$ then $O_m^\nu=D_m^\nu$.
\lambdaabel{LemCm}
\varepsilonnd{Lem}
{\varepsilonm Proof.} If $j'(m,\nu)<m(\nu)$ we have two possibilities regarding $j(m,\nu)$, either $j(m,\nu)<m(\nu)$ or $j(m,\nu)=m(\nu)$. Suppose that $j(m,\nu)<m(\nu)$, then $j(m,\nu)=0$, since $l_{j(m,\nu)}(\nu)<l_{j(m,\nu)+1}(\nu)$ and $l_1(\nu)=\cdots=l_{m(\nu)}(\nu)<l_{m(\nu)+1}(\nu)$. Then $\nu\in H_m$, and by Proposition \rhoef{CHm} we have that $C_m^\nu=V(I^\nu,J_m^\nu)$ where $J_m^\nu=(z^{(0)},\lambdadots,z^{([\pihirac{m}{n}])})$. If $j(m,\nu)=m(\nu)$ we have that $J_m^\nu$
is the ideal
\betaegin{equation}
J_m^\nu=\lambdaeft\{\betaegin{array}{cl}
\lambdaeft(F_{g_1,\nu}^{(0)}\rhoight) & \mbox{ if }c(\nu)=g_1>0\\
\\
\lambdaeft(z^{(0)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle-1)},F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}\rhoight) & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
\lambdaabel{eqJ}
\varepsilonnd{equation}
Therefore the conditions $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$ disappear when taking the Zariski closure.
Suppose now that $m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_2$. We prove that
\betaegin{equation}
\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})}=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})},
\lambdaabel{igD}
\varepsilonnd{equation}
or in other words, the open condition $x_2^{(\nu_2)}\neq 0$ is superfluous when taking the Zariski closure. This claim is obvious for $m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_1$, since $x_2^{(\nu_2)}$ appears in the generators of $J_m^\nu$ at most once (in $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}$ if
$l_{g_1+1}(\nu)\lambdaeq m<l_{g_1+1}(\nu)+e_{g_1+1}$). For $g_1<j'(m,\nu)\lambdaeq g_2$, we are necessarily in the case $g_2=g_1+1$ and we prove the
equality (\rhoef{igD}) by induction on $m$. For $m=l_{g_2}(\nu)+e_{g_2}$, if the equality (\rhoef{igD}) does not hold, then
\[C:=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}}\nsubseteq C_m^\nu.\]
Notice that by Corollary \rhoef{Corolario3}
\[F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}})}={F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}}^{n_{g_2}}-c_{g_2}{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2)}
\cdots {F_{g_2-2,\nu}^{(\pihirac{l_{g_2-1}(\nu)}{e_{g_2-2}})}}^{r_{g_2-1}^{(g_2)}}+G_{g_2,\nu}\]
and by Lemma \rhoef{TechLem} (i) we deduce that, if $x_2^{(\nu_2)}=0$ then $F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}$. Moreover,
since the polynomials $F_\nu^{(l)}$ are quasi-homogeneous
\[F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+1)}\varepsilonquiv\betaar c_{g_2}{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2+1)}\cdots {F_{g_2-2,\nu}^{(\pihirac{l_{g_2-1}(\nu)}{e_{g_2-2}})}}^{r_{g_2-1}^{(g_2)}}\mbox{ mod }x_2^{(\nu_2)}\]
and, by (\rhoef{eqO}), we deduce that $x_2^{(\nu_2+1)}=0$. Hence \[C=V(I^\nu,J_{l_{g_2}(\nu)-1}^\nu,x_2^{(\nu_2)},x_2^{(\nu_2+1)},F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}),\]
and now consider the closed
set $C':=\pii_{m,m-1}(C_m^\nu)=\mathbb overline{V(I^\nu,J_{m-1}^\nu)\cap D(x_1^{(\nu_1)})}$. We have that $\pii_{m,m-1}^{-1}(C')=\mathbb overline{V(I^\nu,
J_m^\nu)\cap D(x_1^{(\nu_1)})}=C_m^\nu\cup C$ with Codim$(C_m^\nu)=\mbox{Codim}(C')+1$ and Codim$(C)=\mbox{Codim}(C')+2$, which is a contradiction. Suppose it true for $m$ and we prove it for $m+1$. Consider $C':=\pii_{m+1,m}(C_m^\nu)=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})}$. By
induction hypothesis $C'=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})}$, and then $\pii_{m+1,m}^{-1}(C')=\mathbb overline{V(I^\nu,J_m^\nu,
F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})}$. If $F_\nu^{(m+1)}= 0$ then we are done. Otherwise, by Corollary \rhoef{Corolario3}, $F_\nu^{(m+1)}={F_{g_2,\nu}^
{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)}}^{e_{g_2}}$ where $r=\pihirac{m+1-l_{g_2}(\nu)}{e_{g_2}}$, and, as in the first step of
induction, if it subdivides as
\[C_{m+1}^\nu\cup \mathbb overline{V(I^\nu,J_m^\nu,F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}},\]
then Codim$(\mathbb overline{V(I^\nu,J_m^\nu,F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}})=\mbox{Codim}(C')+2$ which is a contradiction.
Finally, if $j'(m,\nu)\gammaeq g_2+1$ there is nothing to prove.
\pisifill $\Box$ | 4,024 | 114,405 | en |
train | 0.29.15 | Finally, if $j'(m,\nu)\gammaeq g_2+1$ there is nothing to prove.
\pisifill $\Box$
We will describe a set $F_m\sigmaubset H_m\cup L_m$ such that $\{C_m^\nu\ |\ \nu\in F_m\}$ is the set of irreducible components. The process of
defining $F_m$ as a subset of $H_m\cup L_m$ is done in two steps. The first reduction is easy. We consider the product ordering $\lambdaeq_p$ in $\mathbb Z^2$ given by:
\betaegin{equation}
\nu\lambdaeq_p\nu'\mbox{ if and only if }\nu_i\lambdaeq\nu'_i\mbox{ for }i=1,2.
\lambdaabel{ordenGen}
\varepsilonnd{equation}
\betaegin{Pro}
For $\nu,\nu'\in H_m\cup L_m$ we have that
\betaegin{enumerate}
\item [(i)] If $C_m^{\nu'}\sigmaubseteq C_m^\nu$ then $\nu\lambdaeq_p\nu'$.
\\
\item [(ii)] Moreover if $\nu,\nu'\in H_m\cup L_m^=$ then we have
\[C_m^{\nu'}\sigmaubseteq C_m^\nu\mathbb Longleftrightarrow \nu\lambdaeq_p\nu'.\]
\varepsilonnd{enumerate}
\lambdaabel{Lema2Gen}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
\betaegin{enumerate}
\item [(i)] Suppose that $\nu$ and $\nu'$ are not comparable. Then we can assume that $\nu_1<\nu_1'$ and $\nu_2>\nu_2'$. Then, since $C_m^\nu\sigmaubseteq V(I^\nu)$, and $C_m^{\nu'}\sigmaubset V(I^{\nu'})$, it follows that
\[C_m^\nu\nsubseteq C_m^{\nu'}\mbox{ and }C_m^{\nu'}\nsubseteq C_m^\nu.\]
\item[(ii)] The claim follows by (i) and the definition of $C_m^\nu$ for $\nu\in
H_m\cup L_m^=$.
\varepsilonnd{enumerate}
\pisifill$\Box$
\betaegin{Defi}
According to Proposition \rhoef{Lema2Gen} we define the set:
\[P_m=min_{\lambdaeq_p}\{H_m\cup L_m^=\}.\]
\lambdaabel{defPm}
\varepsilonnd{Defi}
The second reduction, which defines the set $F_m\sigmaubseteq P_m\cup L_m^<$, is much more involved, and the singular locus of the approximated roots play a role now when studying the inclusions $C_m^{\nu'}\sigmaubseteq C_m^\nu$ for different elements $\nu$ and $\nu'$ in $P_m\cup L_m^<$. By Proposition \rhoef{Lema2Gen} (i)
we have to consider $\nu\lambdaneq_p\nu'$, where, by definition of $P_m$, $\nu\in L_m^<$ and $\nu'\in P_m\cup L_m^<$.
\betaegin{Pro}
Given $m\in\mathbb Z_{>0}$, $\nu\in L_m^<$ and $\nu'\in P_m\cup L_m^<$ with $\nu\lambdaneq_p\nu'$, such that $\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}$ then $C_m^{\nu'}\sigmaubseteq C_m^\nu$.
\lambdaabel{PropC1}
\varepsilonnd{Pro}
{\varepsilonm Proof.} We simplify notation by setting $k_i(\nu)=\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}$, for $1\lambdaeq i\lambdaeq g$. By the
description of $\sigma_{Reg,j}$ given in (\rhoef{SigRegj}) we have to prove the inclusion $C_m^{\nu'}\sigmaubseteq C_m^\nu$ when $\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}$ and
$1\lambdaeq j'(m,\nu)\lambdaeq g_2$. Then, by Lemma \rhoef{LemCm}, we have that
\[C_m^\nu=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})}.\]
Suppose first that $\sigma_{Reg,j'(m,\nu)}=\rho_2$, then $\nu'=\nu+(0,\betaeta)$ with $\betaeta>0$. We distinguish two cases:
(i) If $\nu'\in H_m$, then by Proposition \rhoef{CHm},
\[C_m^{\nu'}=V(I^\nu,x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+\betaeta-1)},z^{(0)},\lambdadots,z^{([m/n])}).\]
Note that $g_1=0$, because otherwise $l_1(\nu')=l_1(\nu)$ and $\nu'\notin H_m$. Then, since $1\lambdaeq j'(m,\nu)\lambdaeq g_2$, we deduce that $g_2=1$.
There exists $1\lambdaeq r<k_1(\nu)$ such that
\[l_1(\nu)+re_1\lambdaeq m<l_1(\nu)+(r+1)e_1\]
since $\nu\in L_m^<$. Then
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}\rhoight).\]
Notice that $[\pihirac{m}{n}]=[\pihirac{l_1(\nu)+re_1}{n}]=\lambdaangle\nu,\gammaamma_1\rhoangle+\alphalpha$, where $\alphalpha=[\pihirac{r}{n_1}]$. Now, since $\nu'=\nu+(0,\betaeta)
\in H_m$ and $g_2=1$, we have that $l_1(\nu')=l_1(\nu)+e_1\betaeta\gammaeq m+1$ and it follows that $\betaeta>r$.
Hence we have to prove
\betaegin{equation}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+l)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+\betaeta-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+l-1)})
\lambdaabel{eqqq}
\varepsilonnd{equation}
for $0\lambdaeq l\lambdaeq r$. By Corollary \rhoef{Corolario3} we have $F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})}={z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}x_2^{(\nu_2)}+G_{1,\nu}$.
And by Lemma \rhoef{TechLem} (i), if $x_2^{(\nu_2)}=0$ then $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}=0$, and hence $G_{1,\nu}=0$.
By quasi-homogeneity we can write
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}=c_1{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1-1}z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}+c_2{x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+1)}
x_2^{(\nu_2)}+c_3{x_1^{(\nu_1)}}^{a_1}x_2^{(\nu_2+1)}+G_{1,\nu}^{(1)}\]
where $c_1,c_2,c_3$ are certain coefficients and $G_{1,\nu}^{(1)}$ is a quasi-homogeneous polynomial of degree $\pihirac{l_1(\nu)}{e_1}+1$. We have that $G_{1,\nu}^{(1)}=0$ when $\nu\notin\rho_1\cup\rho_2$, and otherwise, we can apply the same arguments as in the proof of Lemma \rhoef{TechLem} we deduce that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},x_2^{(\nu_2+1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}).\]
Again by quasi-homogeneity, if $n_1<r$,
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)} & =z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+n_1)})+
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}}^{n_1}+x_2^{(\nu_2)}h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\\
\\
& \ x_2^{(\nu_2+1)}h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\cdots+x_2^{(\nu_2+n_1)}h_{2,n_1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+G_{1,\nu}^{(n_1)}.\\
\varepsilonnd{array}\]
And analogously we prove that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+n_1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}).\]
And in general, for $1\lambdaeq k\lambdaeq r$
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+k)} & = z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})+
z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}
h_{1,1}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})\\
\\
& \ +\ \cdots\ +z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])}h_{1,[k/n_1]}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle+k)})+\\
\\
& +h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2)}+h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+1)}+\cdots+\\
\\
& +h_{2,k}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+k)}\\
\varepsilonnd{array}\]
where $h_{1,0},h_{1,1},\lambdadots,h_{1,[k/n_1]},h_{2,0},\lambdadots,h_{2,k}$ are polynomials. And since $k\lambdaeq r<\betaeta$ and $[\pihirac{m}{n}]=\lambdaangle\nu,\gammaamma_1\rhoangle+\alphalpha$
with $\alphalpha=[\pihirac{r}{n_1}]\gammaeq[\pihirac{k}{n_1}]$, we have proved (\rhoef{eqqq}) as we wanted. | 3,296 | 114,405 | en |
train | 0.29.16 | By quasi-homogeneity we can write
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}=c_1{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1-1}z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}+c_2{x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+1)}
x_2^{(\nu_2)}+c_3{x_1^{(\nu_1)}}^{a_1}x_2^{(\nu_2+1)}+G_{1,\nu}^{(1)}\]
where $c_1,c_2,c_3$ are certain coefficients and $G_{1,\nu}^{(1)}$ is a quasi-homogeneous polynomial of degree $\pihirac{l_1(\nu)}{e_1}+1$. We have that $G_{1,\nu}^{(1)}=0$ when $\nu\notin\rho_1\cup\rho_2$, and otherwise, we can apply the same arguments as in the proof of Lemma \rhoef{TechLem} we deduce that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},x_2^{(\nu_2+1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}).\]
Again by quasi-homogeneity, if $n_1<r$,
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)} & =z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+n_1)})+
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}}^{n_1}+x_2^{(\nu_2)}h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\\
\\
& \ x_2^{(\nu_2+1)}h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\cdots+x_2^{(\nu_2+n_1)}h_{2,n_1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+G_{1,\nu}^{(n_1)}.\\
\varepsilonnd{array}\]
And analogously we prove that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+n_1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}).\]
And in general, for $1\lambdaeq k\lambdaeq r$
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+k)} & = z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})+
z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}
h_{1,1}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})\\
\\
& \ +\ \cdots\ +z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])}h_{1,[k/n_1]}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle+k)})+\\
\\
& +h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2)}+h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+1)}+\cdots+\\
\\
& +h_{2,k}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+k)}\\
\varepsilonnd{array}\]
where $h_{1,0},h_{1,1},\lambdadots,h_{1,[k/n_1]},h_{2,0},\lambdadots,h_{2,k}$ are polynomials. And since $k\lambdaeq r<\betaeta$ and $[\pihirac{m}{n}]=\lambdaangle\nu,\gammaamma_1\rhoangle+\alphalpha$
with $\alphalpha=[\pihirac{r}{n_1}]\gammaeq[\pihirac{k}{n_1}]$, we have proved (\rhoef{eqqq}) as we wanted.
(ii) If $\nu'\in L_m$, then by Lemma \rhoef{LemCm}
\[C_m^{\nu'}=\mathbb overline{V(I^{\nu'},J_m^{\nu'})\cap D(x_1^{(\nu_1')})}.\]
Since $\nu\lambdaneq_p\nu'$ we have that $I^\nu\sigmaubseteq I^{\nu'}$. We are going to prove that the generators of
$J_m^\nu$ modulo $(I^{\nu'},z^{(l)})_{0\lambdaeq l<\pihirac{l_1(\nu')}{n}}$ belong to $J_m^{\nu'}$. Since $l_i(\nu)=l_i(\nu')$ for $1\lambdaeq i\lambdaeq g_1$, we have that
\[\betaegin{array}{ll}
F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r_i)}=F_{i,\nu'}^{(\pihirac{l_i(\nu')}{e_i}+r_i)} & \mbox{ for }1\lambdaeq i\lambdaeq g_1-1
\mbox{ and }0\lambdaeq r_i<k_i(\nu)=k_i(\nu'),\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+r)}=F_{g_1,\nu'}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+r)} & \mbox{ for }
0\lambdaeq r<k_{g_1}(\nu)<k_{g_1}(\nu').\\
\varepsilonnd{array}\]
When $J_m^\nu$ has more generators, that is, when $j(m,\nu)>g_1$, then $j'(m,\nu)\gammaeq g_1$. We distinguish two cases.
\betaegin{enumerate}
\item[(ii.a)] If $j'(m,\nu)=g_1$, then, by Corollary \rhoef{Corolario3},
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)};F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r_i)},1\lambdaeq i\lambdaeq g_1,0\lambdaeq r_i<k_i(\nu);F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}\rhoight)_{l_i(\nu)<l_{i+1}(\nu)},\]
and $l_{g_1+1}(\nu)\lambdaeq m<l_{g_1+1}(\nu)+e_{g_1+1}$. By (\rhoef{ecsFs}) we have that
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}= {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}},\]
since $\nu_2'>\nu_2$. Notice that $l_{g_1+1}(\nu)=l_{g_1}(\nu)+l_{g_1+1}(\nu)-l_{g_1}(\nu)=l_{g_1}(\nu')+\alphalpha\lambdaeq m$, with $\alphalpha>0$. Then $l_{g_1}(\nu')<m$ and
therefore $j(m,\nu')\gammaeq g_1$.
If $j(m,\nu')>g_1$ then
\[F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}\in J_m^{\nu'},\]
since $\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}=
\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu)$ and $k_{g_1}(\nu)<k_{g_1}(\nu')$. If $j(m,\nu')=g_1$ then
\[F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}\in J_m^{\nu'},\]
since $\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}=\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu)$, and $k_{g_1}(\nu)\lambdaeq[\pihirac{m-l_{g_1}
(\nu')}{e_{g_1}}]$, because $l_{g_1}(\nu)=l_{g_1}(\nu')$ and $m\gammaeq l_{g_1+1}(\nu)$.
\item[(ii.b)] If $j'(m,\nu)>g_1$, then we are in the case $g_2=g_1+1$ and $j'(m,\nu)=g_1+1$. There exists an integer $1\lambdaeq r<k_{g_1+1}(\nu)$ such that
\betaegin{equation}
l_{g_1+1}(\nu)+re_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu)+(r+1)e_{g_1+1}.
\lambdaabel{toro}
\varepsilonnd{equation}
Then $J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}}{e_{g_1}}+k_{g_1}(\nu)-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+r)}\rhoight)$, where
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}= {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}x_2^{(\nu_2)}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}.\]
And, analogously to case (i), we can write the polynomials $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ for $1\lambdaeq k\lambdaeq r$ as
\[\betaegin{array}{ll}
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)} & =F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}h_{1,0}+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+1)}h_{1,1}+
\cdots+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+
[\pihirac{k}{n_{g_1+1}}])}h_{1,[k/n_{g_1+1}]}+\\
\\
& h_{2,0}x_2^{(\nu_2)}+\cdots+h_{2,k}x_2^{(\nu_2+k)},\\
\varepsilonnd{array}\]
for certain polynomials $h_{1,0},\lambdadots,h_{1,[k/n_{g_1+1}]},h_{2,0},\lambdadots,h_{2,k}$.
If $j(m,\nu')=g_1$ then $l_{g_1+1}(\nu')>m$ and by (\rhoef{toro}) it follows that $\betaeta>r$. Moreover
\[J_m^{\nu'}=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu')}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu')}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}})},\lambdadots,F_{g_1,\nu}^
{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+
\alphalpha)}\rhoight)\]
with $\alphalpha=[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]$.
We have that $[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]=[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}]=[\pihirac{m}{e_{g_1}}]-\pihirac{l_{g_1}(\nu)}{e_{g_1}}$, since $\pihirac{l_{g_1}(\nu)}
{e_{g_1}}$ is an integer, and by (\rhoef{toro}) we have that $\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+\pihirac{r}{n_{g_1+1}}\lambdaeq\pihirac{m}{e_{g_1}}<\lambdaangle\nu,\gammaamma_{g_1+1}
\rhoangle+\pihirac{r+1}{n_{g_1+1}}$. Then $[\pihirac{m}{e_{g_1}}]=\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+[\pihirac{r}{n_{g_1+1}}]$, and we have that
\[\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{k}{n_{g_1+1}}\rhoight]\lambdaeq\pihirac{l_{g_1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}\rhoight].\]
It follows that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ belongs to $J_m^{\nu'}$. | 4,065 | 114,405 | en |
train | 0.29.17 | \item[(ii.b)] If $j'(m,\nu)>g_1$, then we are in the case $g_2=g_1+1$ and $j'(m,\nu)=g_1+1$. There exists an integer $1\lambdaeq r<k_{g_1+1}(\nu)$ such that
\betaegin{equation}
l_{g_1+1}(\nu)+re_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu)+(r+1)e_{g_1+1}.
\lambdaabel{toro}
\varepsilonnd{equation}
Then $J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}}{e_{g_1}}+k_{g_1}(\nu)-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+r)}\rhoight)$, where
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}= {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}x_2^{(\nu_2)}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}.\]
And, analogously to case (i), we can write the polynomials $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ for $1\lambdaeq k\lambdaeq r$ as
\[\betaegin{array}{ll}
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)} & =F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}h_{1,0}+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+1)}h_{1,1}+
\cdots+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+
[\pihirac{k}{n_{g_1+1}}])}h_{1,[k/n_{g_1+1}]}+\\
\\
& h_{2,0}x_2^{(\nu_2)}+\cdots+h_{2,k}x_2^{(\nu_2+k)},\\
\varepsilonnd{array}\]
for certain polynomials $h_{1,0},\lambdadots,h_{1,[k/n_{g_1+1}]},h_{2,0},\lambdadots,h_{2,k}$.
If $j(m,\nu')=g_1$ then $l_{g_1+1}(\nu')>m$ and by (\rhoef{toro}) it follows that $\betaeta>r$. Moreover
\[J_m^{\nu'}=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu')}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu')}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}})},\lambdadots,F_{g_1,\nu}^
{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+
\alphalpha)}\rhoight)\]
with $\alphalpha=[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]$.
We have that $[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]=[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}]=[\pihirac{m}{e_{g_1}}]-\pihirac{l_{g_1}(\nu)}{e_{g_1}}$, since $\pihirac{l_{g_1}(\nu)}
{e_{g_1}}$ is an integer, and by (\rhoef{toro}) we have that $\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+\pihirac{r}{n_{g_1+1}}\lambdaeq\pihirac{m}{e_{g_1}}<\lambdaangle\nu,\gammaamma_{g_1+1}
\rhoangle+\pihirac{r+1}{n_{g_1+1}}$. Then $[\pihirac{m}{e_{g_1}}]=\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+[\pihirac{r}{n_{g_1+1}}]$, and we have that
\[\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{k}{n_{g_1+1}}\rhoight]\lambdaeq\pihirac{l_{g_1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}\rhoight].\]
It follows that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ belongs to $J_m^{\nu'}$.
If $j(m,\nu')=g_1+1$ then $\nu'\in N_{g_+1}$ and therefore $\betaeta=qn_{g_1+1}$ with $q\in\mathbb Z_{>0}$. Moreover there exists an integer $0<r'<k_{g_1+1}(\nu')$ such that
\[l_{g_1+1}(\nu')+r'e_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu')+(r'+1)e_{g_1+1}.\]
Or equivalently $l_{g_1+1}(\nu)+(r'+qn_{g_1+1})e_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu)+(r'+qn_{g_1+1}+1)e_{g_1+1}$. Then by (\rhoef{toro}) it follows that
\[r=r'+qn_{g_1+1},\]
and therefore $r>\betaeta$. Moreover $k_{g_1}(\nu')=k_{g_1}(\nu)+qn_{g_1+1}$. Then
\[J_m^{\nu'}=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu')}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu')}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu')-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}}+r')}\rhoight),\]
where notice that $\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu')-1=\pihirac{l_{g_1}(\nu)}{e_{g_1}}+k_{g_1}(\nu)+qn_{g_1+1}-1$, and
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+k_{g_1}(\nu)-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+r)}\rhoight).\]
Since $I^{\nu'}=(x_1^{(0)},\lambdadots,x_1^{(\nu_1-1)},x_2^{(0)},\lambdadots,x_2^{(\nu_2+qn_{g_1+1}-1)})$ it follows
that for $0\lambdaeq k\lambdaeq r'$ and $s=qn_{g_1+1}+k$
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}= F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}}+k)}.\]
Then finally we have to prove that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}\in J_m^{\nu'}$ for
$0\lambdaeq s<qn_{g_1+1}$. This follows as in the previous cases, since
\[\betaegin{array}{ll}
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & = {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}x_2^{(\nu_2)}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}\\
\\
& \mbox{ and for }0\lambdaeq s<qn_{g_1+1}\\
\\
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)} & = F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}h_{1,0}+\cdots+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+
[\pihirac{s}{n_{g_1+1}}])}h_{1,[s/n_{g_1+1}]}\\
\\
&\ -h_{2,0}x_2^{(\nu_2)} -\cdots-h_{2,s}x_2^{(\nu_2+s)}\\
\varepsilonnd{array}\]
for polynomials $h_{1,0},\lambdadots,h_{1,[s/n_{g_1+1}]},h_{2,0},\lambdadots,h_{2,s}$. And it is clear that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}\in J_m^{\nu'}$.
\varepsilonnd{enumerate}
The key point in all the cases is that $\alphalpha_2^{(g_1+1)}=1$.
If $\nu'-\nu\in\rho_1$ then we are in the case $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Now $\nu_1'>\nu_1$ and $\nu_2'=\nu_2$, and similar arguments apply to this case to get the inclusion we want to prove.
\pisifill$\Box$
The previous Proposition motivates the following definition.
\betaegin{Defi}
We consider the relation in $N_0$, depending on $m$ and denoted by $<_{R,m}$, given by
\[\nu\lambdaeq_{R,m}\nu'\mbox{ if and only if }\nu\lambdaeq_p\nu'\mbox{ and }\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}.\]
We define the set $F_m=\mbox{min}_{\lambdaeq_{R,m}}\{P_m\cup L_m^<\}$.
\lambdaabel{orden2}
\varepsilonnd{Defi}
Notice that, by (\rhoef{SigRegj}), for $m$ and $\nu$ such that $j'(m,\nu)>g_2$, this order is just equality.
It is worth pointing out that the inclusions which are described by this last relation in Proposition \rhoef{PropC1}, can be explained by the fact that even though a curve may be in the singular locus of a quasi-ordinary surface, it may not be part of the singular locus of its first approximated quasi-ordinary surfaces. And as Proposition \rhoef{Cgeom} explains, the geometry of $C_m^\nu$ is only determined by the geometry of one of its approximated roots, for $m$ small enough. Hence, the jets which project to the singular locus of the surface but not to the singular locus of the approximated surfaces will not give rise to irreducible components of the jet schemes for $m$ small enough, and they will be included in other components.
Now we prove that all possible inclusions are controlled by the relation defined in Definition \rhoef{orden2} and the product ordering, that is, in the set $F_m$.
\betaegin{Pro}
Given $m\in\mathbb Z_{>0}$ and $\nu,\nu'\in F_m$ with $\nu\lambdaeq_p\nu'$ then $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$.
\lambdaabel{PropC2}
\varepsilonnd{Pro}
{\varepsilonm Proof.} We will prove that $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$ by showing that
\betaegin{equation}
\mbox{Codim}(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^\nu).
\lambdaabel{eqCodim}
\varepsilonnd{equation}
First notice that $\nu\in L_m^<$, since otherwise there would not exist $\nu'\neq \nu$ such that $\nu'\in F_m$ and $\nu\lambdaeq_p\nu'$. Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$,
we define the set
\[\betaegin{array}{cl}
E(\nu)_m & =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and it is minimal with respect to }\lambdaeq_{R,m}\mbox{ in }\nu+\sigma\}\\
\\
& =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and }\nexists\ \omegaidetilde\nu\in(\nu+\sigma)\cap(P_m\cup L_m^<)\mbox{ such that }\omegaidetilde\nu\lambdaeq_{R,m}\nu'\}.\\
\varepsilonnd{array}\]
We claim that for any $\nu'\in E(\nu)_m$ we have that
\betaegin{equation}
\mbox{Codim}(C_{m_0}^{\nu'})\lambdaeq\mbox{Codim}(C_{m_0}^\nu)\ \mbox{ for }l_1(\nu)+e_1\lambdaeq m_0<l_{i(\nu)}(\nu).
\lambdaabel{eqCodim0}
\varepsilonnd{equation}
We prove this claim by induction on $m$. For $m=l_1(\nu)+e_1$ we have that | 4,082 | 114,405 | en |
train | 0.29.18 | If $\nu'-\nu\in\rho_1$ then we are in the case $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Now $\nu_1'>\nu_1$ and $\nu_2'=\nu_2$, and similar arguments apply to this case to get the inclusion we want to prove.
\pisifill$\Box$
The previous Proposition motivates the following definition.
\betaegin{Defi}
We consider the relation in $N_0$, depending on $m$ and denoted by $<_{R,m}$, given by
\[\nu\lambdaeq_{R,m}\nu'\mbox{ if and only if }\nu\lambdaeq_p\nu'\mbox{ and }\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}.\]
We define the set $F_m=\mbox{min}_{\lambdaeq_{R,m}}\{P_m\cup L_m^<\}$.
\lambdaabel{orden2}
\varepsilonnd{Defi}
Notice that, by (\rhoef{SigRegj}), for $m$ and $\nu$ such that $j'(m,\nu)>g_2$, this order is just equality.
It is worth pointing out that the inclusions which are described by this last relation in Proposition \rhoef{PropC1}, can be explained by the fact that even though a curve may be in the singular locus of a quasi-ordinary surface, it may not be part of the singular locus of its first approximated quasi-ordinary surfaces. And as Proposition \rhoef{Cgeom} explains, the geometry of $C_m^\nu$ is only determined by the geometry of one of its approximated roots, for $m$ small enough. Hence, the jets which project to the singular locus of the surface but not to the singular locus of the approximated surfaces will not give rise to irreducible components of the jet schemes for $m$ small enough, and they will be included in other components.
Now we prove that all possible inclusions are controlled by the relation defined in Definition \rhoef{orden2} and the product ordering, that is, in the set $F_m$.
\betaegin{Pro}
Given $m\in\mathbb Z_{>0}$ and $\nu,\nu'\in F_m$ with $\nu\lambdaeq_p\nu'$ then $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$.
\lambdaabel{PropC2}
\varepsilonnd{Pro}
{\varepsilonm Proof.} We will prove that $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$ by showing that
\betaegin{equation}
\mbox{Codim}(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^\nu).
\lambdaabel{eqCodim}
\varepsilonnd{equation}
First notice that $\nu\in L_m^<$, since otherwise there would not exist $\nu'\neq \nu$ such that $\nu'\in F_m$ and $\nu\lambdaeq_p\nu'$. Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$,
we define the set
\[\betaegin{array}{cl}
E(\nu)_m & =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and it is minimal with respect to }\lambdaeq_{R,m}\mbox{ in }\nu+\sigma\}\\
\\
& =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and }\nexists\ \omegaidetilde\nu\in(\nu+\sigma)\cap(P_m\cup L_m^<)\mbox{ such that }\omegaidetilde\nu\lambdaeq_{R,m}\nu'\}.\\
\varepsilonnd{array}\]
We claim that for any $\nu'\in E(\nu)_m$ we have that
\betaegin{equation}
\mbox{Codim}(C_{m_0}^{\nu'})\lambdaeq\mbox{Codim}(C_{m_0}^\nu)\ \mbox{ for }l_1(\nu)+e_1\lambdaeq m_0<l_{i(\nu)}(\nu).
\lambdaabel{eqCodim0}
\varepsilonnd{equation}
We prove this claim by induction on $m$. For $m=l_1(\nu)+e_1$ we have that
(i) if $a_1=1$ then we are in the case $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $E(\nu)_m=\varepsilonmptyset$, because $\nu+(1,0),\nu+(0,1)\notin N_1$ and $\nu+
(2,0),\nu+(0,2)\in H_m$ but $\nu\lambdaeq_{R,m}\nu+(2,0)$ and $\nu\lambdaeq_{R,m}\nu+(0,2)$.
(ii) If $a_1>1$ then $\nu+(1,0)\in H_m$ and it follows that in fact $E(\nu)_m=\{\nu+(1,0)\}$, because the only other possible $\nu'$ is
\[\betaegin{array}{ll}
\nu'=\nu+(0,1)\in P_m\cup L_m & \mbox{ if }b_1\varepsilonquiv 0\mbox{ mod }n_1\\
\nu'=\nu+(0,2)\in P_m\cup L_m & \mbox{ otherwise}\\
\varepsilonnd{array}\]
and in both cases we have that $\nu\lambdaeq_{R,m}\nu'$. Now, by Lemma \rhoef{Prop1} we have that for $\nu'=\nu+(1,0)$,
\[\mbox{Codim}(C_m^{\nu'})=\nu_1+\nu_2+\pihirac{l_1(\nu)}{n}+2=\mbox{Codim}(C_m^\nu).\]
Suppose that the claim is true for $m-1$ and we prove it for $m$. Let $\nu'$ be an element in $E(\nu)_m$.
(i) If $\nu'\in E(\nu)_{m-1}$, by induction hypothesis, we have that $\mbox{Codim}(C_{m-1}^{\nu'})\lambdaeq\mbox{Codim}(C_{m-1}^\nu)$. By Corollary \rhoef{CorCodim}
we know that, passing from $m-1$ to $m$, the codimension of $C_m^\nu$ grows if and only if $m$ is divisible by $e_{j(m-1,\nu)}$, and it grows by one. But since
$\nu\lambdaeq_p\nu'$ we have that $j(m-1,\nu')\lambdaeq j(m-1,\nu)$ and therefore if $e_{j(m-1,\nu')}$ divides $m$ then $e_{j(m-1,\nu)}$ divides $m$, and it follows that
$\mbox{Codim}(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^\nu)$.
(ii) If $\nu'\notin E(\nu)_{m-1}$, there exists $\omegaidetilde\nu\in E(\nu)_{m-1}$ such that $\omegaidetilde\nu\lambdaeq_{R,m-1}\nu'$ and $\omegaidetilde\nu\not\lambdaeq_{R,m}
\nu'$. By induction hypothesis we have that $\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})\lambdaeq\mbox{Codim}(C_{m-1}^\nu)$, and again, since $\nu\lambdaneq\omegaidetilde\nu$ then
$j(m,\nu)\gammaeq j(m,\omegaidetilde\nu)$ and therefore $\mbox{Codim}(C_m^{\omegaidetilde\nu})\lambdaeq\mbox{Codim}(C_m^\nu)$. Now we are going to prove that $\mbox{Codim}
(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^{\omegaidetilde\nu})$. We have two possibilities, either $\omegaidetilde\nu\in L_m^<$ or $\omegaidetilde\nu\notin L_m^<$.
If $\omegaidetilde\nu\in L_m^<$, then $m=l_{g_2+1}(\omegaidetilde\nu)+e_{g_2+1}$ and
\[\pii_{m,m-1}^{-1}(O_{m-1}^{\omegaidetilde\nu})=V(I^{\omegaidetilde\nu},z^{(0)},\lambdadots,z^{(\lambdaangle\omegaidetilde\nu,\gammaamma_1\rhoangle-1)},F_{1,\omegaidetilde\nu}^
{(\pihirac{l_1(\omegaidetilde\nu)}{e_1})},\lambdadots,F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})},
F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)})\cap D(x_1^{(\omegaidetilde\nu_1)}),\]
where $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})}= {F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}}-{x_1^{(\omegaidetilde\nu_1)}}^
{\alphalpha_1^{(g_2+1)}}{x_2^{(\omegaidetilde\nu_2)}}^{\alphalpha_2^{(g_2+1)}}\cdots F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})}+G_{g_2+1,\omegaidetilde\nu}$, with $\alphalpha_2^{(g_2+1)}>1$, and $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)}= n_{g_2+1}
{F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}-1}F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}}+1)}-x_2^{(\omegaidetilde\nu_2)}H$,
where $H$ is a polynomial in the variables
$H(x_1^{(\omegaidetilde\nu_1)},x_1^{(\omegaidetilde\nu_1+1)},x_2^{(\omegaidetilde\nu_2)},x_2^{(\omegaidetilde\nu_2+1)},\lambdadots,F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})},
F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}}+1)})$. Then
\[(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap
V(x_2^{(\omegaidetilde\nu_2)})}\cup\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap D(x_2^{(\omegaidetilde\nu_2)})},\]
and it is not difficult to see that $(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=C_m^{\nu'}\cup C_m^{\omegaidetilde\nu}$, where $\nu'=\omegaidetilde\nu+(0,\alphalpha)$, with
\[\alphalpha=\lambdaeft\{\betaegin{array}{cl}
1 & \mbox{ if }g_2=g_1\\
\mbox{min}\{n_{g_1+1},k_{g_1+1}(\omegaidetilde\nu)\} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
where remember that $k_i(\omegaidetilde\nu)$ denotes $\pihirac{l_{i+1}(\omegaidetilde\nu)-l_i(\omegaidetilde\nu)}{e_i}$. In both cases we have, by Proposition \rhoef{Prop1}, that $\mbox{Codim}(C_m^{\nu'})=\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})+1=\mbox{Codim}(C_m^{\omegaidetilde\nu})$. | 3,275 | 114,405 | en |
train | 0.29.19 | If $\omegaidetilde\nu\in L_m^<$, then $m=l_{g_2+1}(\omegaidetilde\nu)+e_{g_2+1}$ and
\[\pii_{m,m-1}^{-1}(O_{m-1}^{\omegaidetilde\nu})=V(I^{\omegaidetilde\nu},z^{(0)},\lambdadots,z^{(\lambdaangle\omegaidetilde\nu,\gammaamma_1\rhoangle-1)},F_{1,\omegaidetilde\nu}^
{(\pihirac{l_1(\omegaidetilde\nu)}{e_1})},\lambdadots,F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})},
F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)})\cap D(x_1^{(\omegaidetilde\nu_1)}),\]
where $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})}= {F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}}-{x_1^{(\omegaidetilde\nu_1)}}^
{\alphalpha_1^{(g_2+1)}}{x_2^{(\omegaidetilde\nu_2)}}^{\alphalpha_2^{(g_2+1)}}\cdots F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})}+G_{g_2+1,\omegaidetilde\nu}$, with $\alphalpha_2^{(g_2+1)}>1$, and $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)}= n_{g_2+1}
{F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}-1}F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}}+1)}-x_2^{(\omegaidetilde\nu_2)}H$,
where $H$ is a polynomial in the variables
$H(x_1^{(\omegaidetilde\nu_1)},x_1^{(\omegaidetilde\nu_1+1)},x_2^{(\omegaidetilde\nu_2)},x_2^{(\omegaidetilde\nu_2+1)},\lambdadots,F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})},
F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}}+1)})$. Then
\[(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap
V(x_2^{(\omegaidetilde\nu_2)})}\cup\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap D(x_2^{(\omegaidetilde\nu_2)})},\]
and it is not difficult to see that $(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=C_m^{\nu'}\cup C_m^{\omegaidetilde\nu}$, where $\nu'=\omegaidetilde\nu+(0,\alphalpha)$, with
\[\alphalpha=\lambdaeft\{\betaegin{array}{cl}
1 & \mbox{ if }g_2=g_1\\
\mbox{min}\{n_{g_1+1},k_{g_1+1}(\omegaidetilde\nu)\} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
where remember that $k_i(\omegaidetilde\nu)$ denotes $\pihirac{l_{i+1}(\omegaidetilde\nu)-l_i(\omegaidetilde\nu)}{e_i}$. In both cases we have, by Proposition \rhoef{Prop1}, that $\mbox{Codim}(C_m^{\nu'})=\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})+1=\mbox{Codim}(C_m^{\omegaidetilde\nu})$.
If $\omegaidetilde\nu\notin L_m^<$ then $m=l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)$ with $i(\omegaidetilde\nu)\lambdaeq g_2+1$, since $j'(m-1,\omegaidetilde\nu)\lambdaeq g_2$. We have that
$(\pii_{m,m-1}^{-1}(O_{m-1}^{\omegaidetilde\nu}))_{red}=V(I^{\omegaidetilde\nu},J_{m-1}^{\omegaidetilde\nu},F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac
{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})})\cap D(x_1^{(\omegaidetilde\nu_1)})$, where
\[F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac
{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})}={x_1^{(\omegaidetilde\nu_1)}}^{\alphalpha_1^{(i(\omegaidetilde\nu))}}{x_2^{(\omegaidetilde\nu_2)}}^{\alphalpha_2^
{(i(\omegaidetilde\nu))}} {z^{(\lambdaangle\omegaidetilde\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i(\omegaidetilde\nu))}}\cdots {F_{i(\omegaidetilde\nu)-2,\omegaidetilde\nu}^{(\pihirac{l_{i(\omegaidetilde\nu)-2}
(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)-2}})}}^{r_{i(\omegaidetilde\nu)-1}^{(i(\omegaidetilde\nu))}}+G_{i(\omegaidetilde\nu),\omegaidetilde\nu}.\]
Therefore, by Lemma \rhoef{TechLem}, $F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})}=0$ implies that $ x_2^{(\omegaidetilde
\nu_2)}=0$ because $i(\omegaidetilde\nu)-2<g_2$. And, as before, if $g_2=g_1+1$ and $i(\omegaidetilde\nu)=g_2+1$ then we have that $\nu'=\omegaidetilde\nu+(0,\alphalpha)$ with
$\alphalpha=\mbox{min}\{n_{g_1+1},k_{g_1+1}(\omegaidetilde\nu)\}$. Otherwise $\nu'=\omegaidetilde\nu+(0,1)$, and in both cases we have
\[(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=C_m^{\nu'}\]
with $\mbox{Codim}(C_m^{\nu'})=\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})+1$. Since $\omegaidetilde\nu\in E(\nu)_{m-1}$, it follows that $j(m-1,\nu)>j(m-1,\omegaidetilde\nu)=
i(\omegaidetilde\nu)-1$ and by Corollary \rhoef{CorCodim} we have that $\mbox{Codim}(C_m^\nu)=\mbox{Codim}(C_{m-1}^\nu)+1$, which finishes the proof.
\pisifill $\Box$
Now we can prove the main theorem of this section.
\betaegin{The}
For $m\in\mathbb Z_{>0}$ the decomposition of $\pii_m^{-1}(X_{Sing})$ in irreducible components is given by
\[(\pii_m^{-1}(X_{Sing}))_{red}=\betaigcup_{\nu\in F_m}C_m^\nu.\]
\lambdaabel{TheCaso1}
\varepsilonnd{The}
{\varepsilonm Proof.} The irreducibility of the sets $C_m^\nu$ was proven in Proposition \rhoef{Prop1}. And by Proposition \rhoef{Lema2Gen}, Proposition \rhoef{PropC1} and Proposition \rhoef{PropC2} we have that
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu=\betaigcup_{\nu\in F_m}C_m^\nu.\]
Hence the result follows by Lemma \rhoef{LemTh}.
\pisifill$\Box$
\betaegin{Rem}
For $\nu\in N_g$, if $\nu\in F_{l_g(\nu)}$, then $\nu\in F_m$ for every $m\gammaeq l_g(\nu)$, or in other words, $\nu$ gives rise to an irreducible component for any $m\gammaeq l_g(\nu)$.
\lambdaabel{RemarkFinal}
\varepsilonnd{Rem}
\betaegin{Rem}
When the equisingular dimension is $c=1$ (see Definition \rhoef{equi}), then $g_1=g_2=g$. Moreover we have the following properties for $1\lambdaeq i\lambdaeq g$
\[\betaegin{array}{l}
l_i(\nu)=l_i(\nu+(0,r)),\ \mbox{ for all }r\in\mathbb Z\\
\\
\mbox{if }\nu\in N_i\mbox{ then }\nu+(0,r)\in N_i,\ \mbox{ for all }r\in\mathbb Z\\
\varepsilonnd{array}\]
Hence we deduce that for any $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ we have $\sigma_{Reg,j'(m,\nu)}=\rho_2$, and therefore $F_m=(P_m\cup L_m^<)\cap\rho_1$.
The behaviour of the jet schemes is exactly as the plane curve defined by the Puiseux pairs $\lambdaambda_1^{(1)},\lambdadots,\lambdaambda_g^{(1)}$.
In \cite{Hcur} the second author describes the irreducible components of jets through the origin in the case of plane curves .
\varepsilonnd{Rem}
The previous remark is the simplest evidence of the fact that the irreducible components are only affected by the topological type. This is proved in Theorem \rhoef{topTree}
To any quasi-ordinary surface singularity we can associate a weighted graph, containing information about the irreducible components of jet schemes and how they behave under truncation maps.
\betaegin{Defi}
The weighted graph of the jet schemes of $X$ is the leveled weighted graph $\Gamma$ defined as follows:
\betaegin{itemize}
\item for $m\gammaeq 1$ we represent every irreducible components of $\pii_m^{-1}(X_{Sing})$ by a vertex $V_m$, the sub-index $m$ being the level of the vertex;
\item we join the vertices $V_{m+1}$ and $V_m$ if the canonical morphism $\pii_{m+1,m}$ induces a morphism between the corresponding irreducible components;
\item we weight each vertex by the dimension of the corresponding irreducible component.
\varepsilonnd{itemize}
We define $E\Gamma$ to be the weighted graph that we obtain from $\Gamma$
by weighting any vertex of $\Gamma$ by the embedding dimensions of the corresponding irreducible components (note that by the definition of $\Gamma,$ these vertices are also weighted by their dimensions).
\lambdaabel{defGrafo0}
\varepsilonnd{Defi}
Notice that the data of the codimension together with the embedded dimension permits to distinguish when the vertex corresponds to a hyperplane or a lattice component. Indeed, given a vertex of the graph, let $e$ be the embedded dimension and $c$ the codimension, then the vertex corresponds to a hyperplane component if and only if $e+c=3(m+1)$. Therefore we can extract from $E\Gamma$ a subgraph $\Gamma'$ as follows.
\betaegin{Defi}
We define a weighted subgraph $\Gamma'$ of $E\Gamma$ by adding the condition that we join the vertices $V_m$ (corresponding to a certain component, say $C_m^{\nu'}$) and $V_{m-1}$ (corresponding to $C_m^\nu$) only if
\betaegin{itemize}
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)\lambdaeq g_2$ then $\nu'=\nu+(0,\alphalpha)$ with $\alphalpha$ minimal among the elements in $F_{m}$.
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)>g_2$ then $\nu'=\nu$.
\varepsilonnd{itemize}
\lambdaabel{defGrafo}
\varepsilonnd{Defi}
The important thing about this new graph $\Gamma'$ is that, with the weights, we are able to detect when we pass from a hyperplane component at level $m$ to a lattice component at level $m+1$, as we also do in the graph $E\Gamma$, but now we can follow this component in a unique path in the graph as $m$ grows. This will be useful to prove the following result.
\betaegin{The}
The graph $\Gamma'$ determines and it is determined by the topological type of the singularity.
\lambdaabel{topTree}
\varepsilonnd{The}
{\varepsilonm Proof.} Obviously the graph is determined by the semigroup, and therefore, by \cite{Gau}, by the topological type.
To prove the converse we consider two different sets of generators of the semigroup $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1',\lambdadots,\gammaamma_{g'}'\}$ and we will prove that the corresponding weighted graphs are different too. | 3,723 | 114,405 | en |
train | 0.29.20 | {\varepsilonm Proof.} The irreducibility of the sets $C_m^\nu$ was proven in Proposition \rhoef{Prop1}. And by Proposition \rhoef{Lema2Gen}, Proposition \rhoef{PropC1} and Proposition \rhoef{PropC2} we have that
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu=\betaigcup_{\nu\in F_m}C_m^\nu.\]
Hence the result follows by Lemma \rhoef{LemTh}.
\pisifill$\Box$
\betaegin{Rem}
For $\nu\in N_g$, if $\nu\in F_{l_g(\nu)}$, then $\nu\in F_m$ for every $m\gammaeq l_g(\nu)$, or in other words, $\nu$ gives rise to an irreducible component for any $m\gammaeq l_g(\nu)$.
\lambdaabel{RemarkFinal}
\varepsilonnd{Rem}
\betaegin{Rem}
When the equisingular dimension is $c=1$ (see Definition \rhoef{equi}), then $g_1=g_2=g$. Moreover we have the following properties for $1\lambdaeq i\lambdaeq g$
\[\betaegin{array}{l}
l_i(\nu)=l_i(\nu+(0,r)),\ \mbox{ for all }r\in\mathbb Z\\
\\
\mbox{if }\nu\in N_i\mbox{ then }\nu+(0,r)\in N_i,\ \mbox{ for all }r\in\mathbb Z\\
\varepsilonnd{array}\]
Hence we deduce that for any $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ we have $\sigma_{Reg,j'(m,\nu)}=\rho_2$, and therefore $F_m=(P_m\cup L_m^<)\cap\rho_1$.
The behaviour of the jet schemes is exactly as the plane curve defined by the Puiseux pairs $\lambdaambda_1^{(1)},\lambdadots,\lambdaambda_g^{(1)}$.
In \cite{Hcur} the second author describes the irreducible components of jets through the origin in the case of plane curves .
\varepsilonnd{Rem}
The previous remark is the simplest evidence of the fact that the irreducible components are only affected by the topological type. This is proved in Theorem \rhoef{topTree}
To any quasi-ordinary surface singularity we can associate a weighted graph, containing information about the irreducible components of jet schemes and how they behave under truncation maps.
\betaegin{Defi}
The weighted graph of the jet schemes of $X$ is the leveled weighted graph $\Gamma$ defined as follows:
\betaegin{itemize}
\item for $m\gammaeq 1$ we represent every irreducible components of $\pii_m^{-1}(X_{Sing})$ by a vertex $V_m$, the sub-index $m$ being the level of the vertex;
\item we join the vertices $V_{m+1}$ and $V_m$ if the canonical morphism $\pii_{m+1,m}$ induces a morphism between the corresponding irreducible components;
\item we weight each vertex by the dimension of the corresponding irreducible component.
\varepsilonnd{itemize}
We define $E\Gamma$ to be the weighted graph that we obtain from $\Gamma$
by weighting any vertex of $\Gamma$ by the embedding dimensions of the corresponding irreducible components (note that by the definition of $\Gamma,$ these vertices are also weighted by their dimensions).
\lambdaabel{defGrafo0}
\varepsilonnd{Defi}
Notice that the data of the codimension together with the embedded dimension permits to distinguish when the vertex corresponds to a hyperplane or a lattice component. Indeed, given a vertex of the graph, let $e$ be the embedded dimension and $c$ the codimension, then the vertex corresponds to a hyperplane component if and only if $e+c=3(m+1)$. Therefore we can extract from $E\Gamma$ a subgraph $\Gamma'$ as follows.
\betaegin{Defi}
We define a weighted subgraph $\Gamma'$ of $E\Gamma$ by adding the condition that we join the vertices $V_m$ (corresponding to a certain component, say $C_m^{\nu'}$) and $V_{m-1}$ (corresponding to $C_m^\nu$) only if
\betaegin{itemize}
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)\lambdaeq g_2$ then $\nu'=\nu+(0,\alphalpha)$ with $\alphalpha$ minimal among the elements in $F_{m}$.
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)>g_2$ then $\nu'=\nu$.
\varepsilonnd{itemize}
\lambdaabel{defGrafo}
\varepsilonnd{Defi}
The important thing about this new graph $\Gamma'$ is that, with the weights, we are able to detect when we pass from a hyperplane component at level $m$ to a lattice component at level $m+1$, as we also do in the graph $E\Gamma$, but now we can follow this component in a unique path in the graph as $m$ grows. This will be useful to prove the following result.
\betaegin{The}
The graph $\Gamma'$ determines and it is determined by the topological type of the singularity.
\lambdaabel{topTree}
\varepsilonnd{The}
{\varepsilonm Proof.} Obviously the graph is determined by the semigroup, and therefore, by \cite{Gau}, by the topological type.
To prove the converse we consider two different sets of generators of the semigroup $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1',\lambdadots,\gammaamma_{g'}'\}$ and we will prove that the corresponding weighted graphs are different too.
Given a weighted graph we can recover the number of characteristic exponents in the following way. Any vertex $V_m$ on the graph comes with the codimension $c(V_m)$ and the embedded dimension $e(V_m)$. Take an infinite branch, and consider the finite part that starts at
\[m_0=\mbox{max }\{m\ |\ V_{m-1}\mbox{ is a hyperplane component and }V_m\mbox{ is a lattice component}\},\]
and ends at
\[m_1=\mbox{min }\{m\ |\ c(V_m)=c(V_{m-1})+1\mbox{ for all }m>m_1\}.\]
We can read $e_0,\lambdadots,e_{g-1}$ making use of Corollary \rhoef{CorCodim}. Indeed, along the piece of branch, the vertex $V_m$ corresponds to a component $C_m^\nu$ with $\nu\in N_g$, $m_0=l_1(\nu)$ and $m_1=l_g(\nu)$. To read this data we consider only branches that projects into the component $Z_1$ of the singular locus, since otherwise we can only assure that $m_0=l_{m(\nu)}(\nu)$, and we do not have all the information whenever $m(\nu)>1$. Notice that $Z_1$ is always a component of the singular locus unless we are in the case $g=1$ and $\gammaamma=(\pihirac{1}{n},\pihirac{1}{n})$, which is very easy to recognize. Indeed, it is the only case when at level $m=1$ we have only one component, with codimension 3 and embedded dimension 0. Moreover the multiplicity $n$ equals the first time $m$ when we have a lattice component. Therefore this simple case is very easily understood in the graph. For the rest of the cases, since we know that $\nu\in N_g$, going backwards we look for the biggest $m'$ such that $c(V_{m'})=c(V_{m_0})-1$. Then $n=m_0-m'$. Now, going from level $m_0$ to $m_1$, we know that the codimension grows by one exactly every $e_1$ steps at first, after every $e_2$ steps, and so on. Since $e_1>e_2>\cdots>e_g=1$ we can read these numbers on the graph. Notice that equivalently we get $n_1,\lambdadots,n_g$, and in particular we have $g$, the number of characteristic exponents.
Suppose now that the number of generators of the semigroups is the same, say $g$. We will prove by induction on $g$ that the graphs corresponding to different sets of generators, are different. We denote the vertices at level $m$ by $V_m(c(V_m),e(V_m))$. For $g=1$, the multiplicity is read from the graph as was explained before, and the situation for $m=1$ is:
\[\betaegin{array}{cl}
\betaullet\ V_1(3,0) & \mbox{ if }\gammaamma=(\pihirac{1}{n},\pihirac{1}{n})\\
\\
\betaullet\ V_1(2,1) & \mbox{ if }\gammaamma=(\pihirac{a}{n},\pihirac{1}{n}),\mbox{ with }a>1\\
\\
\betaullet\ V_1(2,1)\ \ \betaullet\ V_1(2,1) & \mbox{ if }\gammaamma=(\pihirac{a}{n},\pihirac{b}{n}),\mbox{ with }b>1\\
\varepsilonnd{array}\]
If we want to compare the graph associated to $\gammaamma$ and the graph associated to $\gammaamma'$, we just have to consider the cases $\gammaamma=(\pihirac{a}{n},\pihirac{1}{n})$, $\gammaamma'=(\pihirac{a'}{n},\pihirac{1}{n})$ with $a\neq a'$, and $\gammaamma=(\pihirac{a}{n},\pihirac{b}{n})$, $\gammaamma'=(\pihirac{a'}{n},\pihirac{b'}{n})$ with $b,b'\neq 1$ and $\gammaamma\neq\gammaamma'$. The first case is very easy to distinguish, since the first moment a component splits in two is at $m=a$ for one graph, and at $m=a'$ for the other. For the other case, first note that the graph of any quasi-ordinary with only one characteristic exponent $\gammaamma=(\pihirac{a}{n},\pihirac{b}{n})$ is the graph associated to $z^n-x_1^ax_2^b$. The key point is that, when $b>1$ we have in the graph a branch which generically corresponds to $x_2^{(0)}\neq 0$ (resp. $x_1^{(0)}\neq 0$), that is, it behaves like the graph of the curve $z^n-x_1^a$ (resp. $z^n-x_2^b$). Therefore comparing graphs associated to $\gammaamma$ and $\gammaamma'$ with $\gammaamma\neq\gammaamma'$, we deduce from Theorem 3.3 in \cite{Hcur}, that the graphs must be different.
Now, suppose it is true for $g-1$ characteristic exponents, and we will prove it for $g$. From Proposition \rhoef{Cgeom} we deduce that is sufficient to prove that the graphs associated to the sets $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$ are different, since otherwise it holds by induction hypothesis. Moreover, since we read the integers $n_1,\lambdadots,n_g$ in the graph, we assume that $n_g'=n_g$. As in the case $g=1$, by looking at the singular locus (which is seen at $m=1$) we just have to consider the case
$\gammaamma_g^{(2)}=\gammaamma_g'^{(2)}=\pihirac{1}{n_g}$ and the case $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$. In the first case $\gammaamma_g^{(1)}\neq\gammaamma_g'^{(1)}$ and $\gammaamma_i^{(2)}=\gammaamma_i'^{(2)}=0$ for $1\lambdaeq i\lambdaeq g-1$. Therefore the graphs are the same till we get to level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$, where $\nu=(\nu_1,0)\in\sigma_{Sing}\cap N_{g-1}$ with $\nu_1$ smallest with this property. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ the graphs must differ at some moment.
Finally, when $\gammaamma\neq\gammaamma'$ with $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$, again by Proposition \rhoef{Cgeom}, the graphs must be the same for $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$, till the last approximated root, that is, $f$, starts playing a role in the definition of a component, say $C^\nu$. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ we will see the difference on the graphs at level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$.
\pisifill$\Box$
\sigmaubsection{Log-canonical threshold}
In \cite{Mus}, Musta\c ta gave a formula of the log-canonical threshold in terms of the codimension of jet schemes, which in our setting can be stated as
\betaegin{equation}
lct(f)=\mbox{min}_{m\gammaeq 0}\pihirac{\mbox{Codim}(X_m)}{m+1}.
\lambdaabel{lct}
\varepsilonnd{equation}
Then, as an application to Theorem \rhoef{TheCaso1}, we can recover, for the case of surfaces, the result in \cite{BGG}.
\betaegin{Cor}
The log-canonical threshold of a quasi-ordinary surface singularity is given by:
\[lct_0(X,\mathbb A^3)=\lambdaeft\{\betaegin{array}{cl}
\pihirac{1+\lambda_1^{(1)}}{e_0\lambda_1^{(1)}} & \ \mbox{ if }\lambda_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\\
\\
1 & \mbox{ if }\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g=1\\
\\
\pihirac{n_1(1+\lambda_2^{(1)})}{e_1(n_1(1+\lambda_2^{(1)})-1)} & \ \mbox{ if }\lambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g>1\\
\varepsilonnd{array}\rhoight.\]
Moreover, the components that contribute to the log canonical threshold are
\[\betaegin{array}{ll}
C_{l_1(\nu)-1}^\nu & \mbox{ if }\gammaamma_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ or }g=1\\
\\
C_{l_2(\nu)-1}^\nu & \mbox{ otherwise}\\
\varepsilonnd{array}\]
where $\nu=(l,0)\in N_1$ if $\gammaamma_1\neq (\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $\nu=(l,0)\in N_2$ otherwise.
\lambdaabel{Thlct}
\varepsilonnd{Cor} | 3,952 | 114,405 | en |
train | 0.29.21 | Now, suppose it is true for $g-1$ characteristic exponents, and we will prove it for $g$. From Proposition \rhoef{Cgeom} we deduce that is sufficient to prove that the graphs associated to the sets $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$ are different, since otherwise it holds by induction hypothesis. Moreover, since we read the integers $n_1,\lambdadots,n_g$ in the graph, we assume that $n_g'=n_g$. As in the case $g=1$, by looking at the singular locus (which is seen at $m=1$) we just have to consider the case
$\gammaamma_g^{(2)}=\gammaamma_g'^{(2)}=\pihirac{1}{n_g}$ and the case $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$. In the first case $\gammaamma_g^{(1)}\neq\gammaamma_g'^{(1)}$ and $\gammaamma_i^{(2)}=\gammaamma_i'^{(2)}=0$ for $1\lambdaeq i\lambdaeq g-1$. Therefore the graphs are the same till we get to level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$, where $\nu=(\nu_1,0)\in\sigma_{Sing}\cap N_{g-1}$ with $\nu_1$ smallest with this property. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ the graphs must differ at some moment.
Finally, when $\gammaamma\neq\gammaamma'$ with $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$, again by Proposition \rhoef{Cgeom}, the graphs must be the same for $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$, till the last approximated root, that is, $f$, starts playing a role in the definition of a component, say $C^\nu$. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ we will see the difference on the graphs at level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$.
\pisifill$\Box$
\sigmaubsection{Log-canonical threshold}
In \cite{Mus}, Musta\c ta gave a formula of the log-canonical threshold in terms of the codimension of jet schemes, which in our setting can be stated as
\betaegin{equation}
lct(f)=\mbox{min}_{m\gammaeq 0}\pihirac{\mbox{Codim}(X_m)}{m+1}.
\lambdaabel{lct}
\varepsilonnd{equation}
Then, as an application to Theorem \rhoef{TheCaso1}, we can recover, for the case of surfaces, the result in \cite{BGG}.
\betaegin{Cor}
The log-canonical threshold of a quasi-ordinary surface singularity is given by:
\[lct_0(X,\mathbb A^3)=\lambdaeft\{\betaegin{array}{cl}
\pihirac{1+\lambda_1^{(1)}}{e_0\lambda_1^{(1)}} & \ \mbox{ if }\lambda_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\\
\\
1 & \mbox{ if }\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g=1\\
\\
\pihirac{n_1(1+\lambda_2^{(1)})}{e_1(n_1(1+\lambda_2^{(1)})-1)} & \ \mbox{ if }\lambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g>1\\
\varepsilonnd{array}\rhoight.\]
Moreover, the components that contribute to the log canonical threshold are
\[\betaegin{array}{ll}
C_{l_1(\nu)-1}^\nu & \mbox{ if }\gammaamma_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ or }g=1\\
\\
C_{l_2(\nu)-1}^\nu & \mbox{ otherwise}\\
\varepsilonnd{array}\]
where $\nu=(l,0)\in N_1$ if $\gammaamma_1\neq (\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $\nu=(l,0)\in N_2$ otherwise.
\lambdaabel{Thlct}
\varepsilonnd{Cor}
{\varepsilonm Proof.}
The case $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g=1$ behaves as an $A_n$-singularity, and then $lct(f)=1$. For the rest of the cases, by Corollary \rhoef{CorCodim}, the codimension of a component grows faster as $m$ grows, for
bigger $j(m,\nu)$. Therefore, the smaller codimension will be attached for $\nu\in P_m\cap F_m$, and more concretely for
$\nu\in H_m\cap F_m$ whenever $H_m\cap F_m\neq\varepsilonmptyset$. If $g_1=0$, since $a_1\gammaeq b_1$, we deduce that the minimal codimension among the elements in $P_m\cap F_m$
is attached for $\nu$ of the form $\nu=(l,0)$, while if $g_1>0$ then $P_m\cap F_m$ consists of just a point of the form $\nu=(l,0)$.
We want to minimize not just the codimension, but the quotient $\pihirac{\mbox{Codim}(X_m)}{m+1}$. That is, to find the biggest $m$ such that $\nu$ still
belongs to $P_m\cap F_m$. Then, when the first characteristic exponent is different from $(\pihirac{1}{n_1},\pihirac{1}{n_1})$, this is
attached for $m=l_1(\nu)-1$ such that $\nu\in L_{m+1}^=$. Then $m=l_1(l,0)-1$ and Codim$(C_m^\nu)=l+[\pihirac{m}{n}]+1$, and since $\nu\in L_{m+1}^=$, $(l,0)\in N_1$
and therefore Codim$(C_m^\nu)=l+l\pihirac{a_1}{n_1}$, which implies that $\pihirac{\mbox{Codim}(C_m^\nu)}{m+1}=\pihirac{a_1+n_1}{na_1}$.
If $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g>1$, what happens is that when $m=l_1(\nu)$ there is no subdivision of the component and $\sigma_{Reg,1}=\rho_1\cup\rho_2$.
If we denote the second exponent by $\gammaamma_2=(\pihirac{\alphalpha_2}{n_1n_2},\pihirac{\betaeta_2}{n_1n_2})$, we look for $\nu$ of the form $(l,0)$ such that $m+1=l_2(\nu)$
with $\nu\in N_2$. Then $\mbox{Codim}(C_m^\nu)=l+\pihirac{l_1(\nu)}{n}+[\pihirac{m-l_1(\nu)}{e_1}]+1=l+\pihirac{l_1(\nu)}{n}+\pihirac{l_2(\nu)-l_1(\nu)}{e_1}$, and therefore
$\pihirac{\mbox{Codim}(C_m^\nu)}{m+1}=\pihirac{l+l\pihirac{1}{n_1}+\pihirac{1}{e_1}(e_2n_2l\pihirac{\alphalpha_2}{n_1n_2}-e_1n_1l\pihirac{1}{n_1})}{e_2n_2l\pihirac{\alphalpha_2}{n_1n_2}}=
\pihirac{1+\pihirac{1}{n_1}+\pihirac{\alphalpha_2}{n_1n_2}-1}{e_2\pihirac{\alphalpha_2}{n_1}}=\pihirac{1+\pihirac{\alphalpha_2}{n_2}}{e_1\pihirac{\alphalpha_2}{n_2}}$.
This coincides with the statement since $\lambda_2=(\pihirac{\alphalpha_2}{n_1n_2}-\pihirac{n_1-1}{n_1},\pihirac{\betaeta_2}{n_1n_2}-\pihirac{n_1-1}{n_1})$.
\pisifill $\Box$
\betaegin{Rem}
Notice that $\pihirac{1+\lambdaambda_1^{(1)}}{e_0\lambdaambda_1^{(1)}}\lambdaeq 1$ except in the case $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g=1$. Moreover in the case of
surfaces the condition $\lambda_1^{(1)}=\pihirac{1}{n_1}$ is equivalent to $\lambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ since the branch is normalized. Then, see notations in
\cite{BGG}, $\varepsilonll_1=\varepsilonll_2$ and in Theorem \rhoef{Thlct} we recover, for the case of surfaces, the formula given in Theorem 3.1 in \cite{BGG}.
\varepsilonnd{Rem}
We now deduce a family of examples whose log canonical threshold can not be computed by a monomial valuation.
\betaegin{Cor}
Let $X$ be a quasi-ordinary surface singularity with $g>1$ characteristic exponents, and such that $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Then $lct(X,\mathbb A^3)$ can not be contributed by monomial valuations in
any variables.
\lambdaabel{CorMV}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It follows from Corollary \rhoef{Thlct} that $lct(X,\mathbb A^3)$ is contributed by $C_{l_1(\nu)}^\nu$, for $\nu$ as is made precise in the above statement.
This is equivalent to say that the valuation
\[\betaegin{array}{ll}
\mathcal V_{C_{l_2(\nu)-1}^\nu} :& \mathbb C[[x_1,x_2,z]]\lambdaongrightarrow\mathbb N\\
\\
& \ \ \ \ \ \ h\ \lambdaongmapsto\ \mbox{ord}_t(h\circ\varepsilonta)\\
\varepsilonnd{array}\]
where $\varepsilonta$ is the generic point of $({\mathbb P}si_{l_2(\nu)-1}^{\mathbb A^3})^{-1}(C_{l_2(\nu)-1}^\nu)$. Note that $\nu$ can take all the values described in Corollary \rhoef{Thlct} but since ${z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-x_1^{(\nu_1)}x_2^{(\nu_2)}=0$ is one of the defining equations of $C_{l_2(\nu)-1}^\nu$, then $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>n_1\mathcal V_{C_{l_2(\nu)-1}^\nu}(z)$ and $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_1)+\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_2)$. Therefore $\mathcal V_{C_{l_2(\nu)-1}^\nu}$ is not a monomial valuation.
\pisifill$\Box$
\sigmaubsection{Example}
\lambdaabel{exs}
Consider the quasi-ordinary surface $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$. The semigroup is generated by the vectors
\[\gammaamma_1=\lambdaeft(\pihirac{3}{2},0\rhoight),\ \gammaamma_2=\lambdaeft(\pihirac{7}{2},\pihirac{3}{2}\rhoight)\mbox{ and }\gammaamma_3=\lambdaeft(\pihirac{29}{4},\pihirac{13}{4}\rhoight).\]
We have that $g_1=g_2=1$.
The singular locus is reducible, of the form
\[X_{Sing}=\{z=x_1=0\}\cup\{x_2=z^2-x_1^3=0\}=Z_1\cup Z_2.\]
Then $\sigma_{Sing}=\mathbb R_{\gammaeq 0}^2\sigmaetminus\{0\}$ and $\sigma_{Reg,1}=\rho_2,\ \sigma_{Reg,2}=\sigma_{Reg,3}=\{(0,0)\}$.
The set $F_m$ describing the irreducible components is the following, for some $m$:
\[\betaegin{array}{l}
F_m=\{(1,0),(0,1)\}, \ \mbox{ for }1\lambdaeq m<6\\
\\
F_m=\{(1,0),(0,2)\}, \ \mbox{ for }6\lambdaeq m<12\\
\\
F_{12}=\{(2,0),(0,2)\}\\
\\
F_{13}=\{(2,0),(0,3)\} \\
\\
F_{18}=\{(2,0),(0,4)\} \\
\\
F_{26}=\{(2,0),(0,4),(0,5)\} \\
\\
F_{28}=\{(3,0),(2,0),(0,4),(0,5)\} \\
\varepsilonnd{array}\]
and the result can be checked by lifting the components $Z_1$ and $Z_2$ of the singular locus to level $m$ as the following graph shows (we did not draw the weights of the vertices for clearness).
\betaegin{figure}[h]
\unitlength=1mm
\betaegin{center}
\betaegin{picture}(110,100)(0,3)
\lambdainethickness{0.15mm}
\piut(24,0){$Z_1$}\piut(80,0){$Z_2$}
\piut(6,4){$m=1$}\piut(25,4){$\betaullet$}\piut(80,4){$\betaullet$}
\piut(26,5){\lambdaine(0,1){5}}\piut(81,5){\lambdaine(0,1){5}}
\piut(6,9){$m=2$}\piut(25,9){$\betaullet$}\piut(80,9){$\betaullet$}
\piut(11,13){$\vdots$}\piut(25,13){$\vdots$}\piut(80,13){$\vdots$}
\piut(6,20){$m=25$}\piut(25,20){$\betaullet$}\piut(80,20){$\betaullet$}
\piut(26,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(1,1){5}} | 4,039 | 114,405 | en |
train | 0.29.22 | \betaegin{Cor}
Let $X$ be a quasi-ordinary surface singularity with $g>1$ characteristic exponents, and such that $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Then $lct(X,\mathbb A^3)$ can not be contributed by monomial valuations in
any variables.
\lambdaabel{CorMV}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It follows from Corollary \rhoef{Thlct} that $lct(X,\mathbb A^3)$ is contributed by $C_{l_1(\nu)}^\nu$, for $\nu$ as is made precise in the above statement.
This is equivalent to say that the valuation
\[\betaegin{array}{ll}
\mathcal V_{C_{l_2(\nu)-1}^\nu} :& \mathbb C[[x_1,x_2,z]]\lambdaongrightarrow\mathbb N\\
\\
& \ \ \ \ \ \ h\ \lambdaongmapsto\ \mbox{ord}_t(h\circ\varepsilonta)\\
\varepsilonnd{array}\]
where $\varepsilonta$ is the generic point of $({\mathbb P}si_{l_2(\nu)-1}^{\mathbb A^3})^{-1}(C_{l_2(\nu)-1}^\nu)$. Note that $\nu$ can take all the values described in Corollary \rhoef{Thlct} but since ${z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-x_1^{(\nu_1)}x_2^{(\nu_2)}=0$ is one of the defining equations of $C_{l_2(\nu)-1}^\nu$, then $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>n_1\mathcal V_{C_{l_2(\nu)-1}^\nu}(z)$ and $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_1)+\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_2)$. Therefore $\mathcal V_{C_{l_2(\nu)-1}^\nu}$ is not a monomial valuation.
\pisifill$\Box$
\sigmaubsection{Example}
\lambdaabel{exs}
Consider the quasi-ordinary surface $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$. The semigroup is generated by the vectors
\[\gammaamma_1=\lambdaeft(\pihirac{3}{2},0\rhoight),\ \gammaamma_2=\lambdaeft(\pihirac{7}{2},\pihirac{3}{2}\rhoight)\mbox{ and }\gammaamma_3=\lambdaeft(\pihirac{29}{4},\pihirac{13}{4}\rhoight).\]
We have that $g_1=g_2=1$.
The singular locus is reducible, of the form
\[X_{Sing}=\{z=x_1=0\}\cup\{x_2=z^2-x_1^3=0\}=Z_1\cup Z_2.\]
Then $\sigma_{Sing}=\mathbb R_{\gammaeq 0}^2\sigmaetminus\{0\}$ and $\sigma_{Reg,1}=\rho_2,\ \sigma_{Reg,2}=\sigma_{Reg,3}=\{(0,0)\}$.
The set $F_m$ describing the irreducible components is the following, for some $m$:
\[\betaegin{array}{l}
F_m=\{(1,0),(0,1)\}, \ \mbox{ for }1\lambdaeq m<6\\
\\
F_m=\{(1,0),(0,2)\}, \ \mbox{ for }6\lambdaeq m<12\\
\\
F_{12}=\{(2,0),(0,2)\}\\
\\
F_{13}=\{(2,0),(0,3)\} \\
\\
F_{18}=\{(2,0),(0,4)\} \\
\\
F_{26}=\{(2,0),(0,4),(0,5)\} \\
\\
F_{28}=\{(3,0),(2,0),(0,4),(0,5)\} \\
\varepsilonnd{array}\]
and the result can be checked by lifting the components $Z_1$ and $Z_2$ of the singular locus to level $m$ as the following graph shows (we did not draw the weights of the vertices for clearness).
\betaegin{figure}[h]
\unitlength=1mm
\betaegin{center}
\betaegin{picture}(110,100)(0,3)
\lambdainethickness{0.15mm}
\piut(24,0){$Z_1$}\piut(80,0){$Z_2$}
\piut(6,4){$m=1$}\piut(25,4){$\betaullet$}\piut(80,4){$\betaullet$}
\piut(26,5){\lambdaine(0,1){5}}\piut(81,5){\lambdaine(0,1){5}}
\piut(6,9){$m=2$}\piut(25,9){$\betaullet$}\piut(80,9){$\betaullet$}
\piut(11,13){$\vdots$}\piut(25,13){$\vdots$}\piut(80,13){$\vdots$}
\piut(6,20){$m=25$}\piut(25,20){$\betaullet$}\piut(80,20){$\betaullet$}
\piut(26,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(1,1){5}}
\piut(6,25){$m=26$}\piut(25,25){$\betaullet$}\piut(80,25){$\betaullet$}\piut(85,25){$\betaullet$}
\piut(26,26){\lambdaine(0,1){5}}\piut(81,26){\lambdaine(0,1){5}}
\piut(6,30){$m=27$}\piut(25,30){$\betaullet$}\piut(80,30){$\betaullet$}\piut(86,26){\lambdaine(1,1){5}}\piut(90,31){$\nearrow$}
\piut(26,31){\lambdaine(0,1){5}}\piut(26,31){\lambdaine(4,1){20}}\piut(81,31){\lambdaine(0,1){5}}
\piut(6,35){$m=28$}\piut(25,35){$\betaullet$}\piut(45,35){$\betaullet$}\piut(80,35){$\betaullet$}
\piut(11,39){$\vdots$}\piut(25,39){$\vdots$}\piut(45,39){$\vdots$}\piut(80,39){$\vdots$}
\piut(6,45){$m=37$}\piut(25,45){$\betaullet$}\piut(45,45){$\betaullet$}\piut(80,45){$\betaullet$}
\piut(26,46){\lambdaine(0,1){5}}\piut(46,46){\lambdaine(0,1){5}}\piut(81,46){\lambdaine(0,1){5}}\piut(81,46){\lambdaine(1,1){5}}
\piut(6,50){$m=38$}\piut(25,50){$\betaullet$}\piut(45,50){$\betaullet$}\piut(80,50){$\betaullet$}\piut(85,50){$\betaullet$}
\piut(26,51){\lambdaine(0,1){5}}\piut(46,51){\lambdaine(0,1){5}}\piut(81,51){\lambdaine(0,1){5}}
\piut(6,55){$m=39$}\piut(25,55){$\betaullet$}\piut(45,55){$\betaullet$}\piut(80,55){$\betaullet$}
\piut(26,56){\lambdaine(0,1){5}}\piut(46,56){\lambdaine(0,1){5}}\piut(81,56){\lambdaine(0,1){5}}
\piut(6,60){$m=40$}\piut(25,60){$\betaullet$}\piut(45,60){$\betaullet$}\piut(80,60){$\betaullet$}
\piut(26,61){\lambdaine(0,1){5}}\piut(46,61){\lambdaine(0,1){5}}\piut(81,61){\lambdaine(0,1){5}}
\piut(6,65){$m=41$}\piut(25,65){$\betaullet$}\piut(45,65){$\betaullet$}\piut(80,65){$\betaullet$}
\piut(26,66){\lambdaine(0,1){5}}\piut(46,66){\lambdaine(0,1){5}}\piut(46,67){$\nearrow$}\piut(81,66){\lambdaine(0,1){5}}
\piut(6,70){$m=42$}\piut(25,70){$\betaullet$}\piut(45,70){$\betaullet$}\piut(80,70){$\betaullet$}
\piut(11,74){$\vdots$}\piut(25,74){$\vdots$}\piut(45,74){$\vdots$}\piut(80,74){$\vdots$}
\piut(6,80){$m=49$}\piut(25,80){$\betaullet$}\piut(45,80){$\betaullet$}\piut(80,80){$\betaullet$}
\piut(26,81){\lambdaine(0,1){5}}\piut(46,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(1,1){5}}
\piut(6,85){$m=50$}\piut(25,85){$\betaullet$}\piut(45,85){$\betaullet$}\piut(80,85){$\betaullet$}\piut(85,85){$\betaullet$}
\piut(26,86){\lambdaine(0,1){5}}\piut(46,86){\lambdaine(0,1){5}}\piut(81,86){\lambdaine(0,1){5}}\piut(86,86){\lambdaine(0,1){5}}
\piut(6,90){$m=51$}\piut(25,90){$\betaullet$}\piut(45,90){$\betaullet$}\piut(80,90){$\betaullet$}\piut(85,90){$\betaullet$}
\piut(26,91){\lambdaine(0,1){5}}\piut(26,91){\lambdaine(1,1){5}}\piut(46,91){\lambdaine(0,1){5}}\piut(81,91){\lambdaine(0,1){5}}\piut(86,91){\lambdaine(0,1){5}}
\piut(6,95){$m=52$}\piut(25,95){$\betaullet$}\piut(30,95){$\betaullet$}\piut(45,95){$\betaullet$}\piut(80,95){$\betaullet$}\piut(85,95){$\betaullet$}
\piut(26,96){\lambdaine(0,1){5}}\piut(31,96){\lambdaine(0,1){5}}\piut(46,96){\lambdaine(0,1){5}}\piut(81,96){\lambdaine(0,1){5}}\piut(86,97){$\nearrow$}
\piut(6,100){$m=53$}\piut(25,100){$\betaullet$}\piut(30,100){$\betaullet$}\piut(45,100){$\betaullet$}\piut(80,100){$\betaullet$}
\varepsilonnd{picture}
\varepsilonnd{center}
\caption{The graph of the surface defined by $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$.}
\lambdaabel{figEx1}
\varepsilonnd{figure}
The arrows in the figure represent the behaviour explained in Remark \rhoef{RemarkFinal}.
Now we give some explanations to illustrate how Proposition \rhoef{Lema2Gen} and Proposition \rhoef{PropC1} work.
For $m=1$, we have $H_1=\{(1,0),(0,1)\}$, $L_1^=\{(0,1)\}$ and $L_1^<=\varepsilonmptyset$. The claim on $F_1$ in this case follows easily by Proposition \rhoef{Lema2Gen}.
At level $m=6$ we have $H_6=\{\nu\in[0,m]^2\cap N_0\ |\ \nu_1\neq 0\}$, $L_6^==\varepsilonmptyset$ and $L_6^<=\{(0,\nu_2)\ |\ 2\lambdaeq\nu_2\lambdaeq 6\}$ with $j'(6,(0,2))=1$, then, by Proposition \rhoef{PropC1} $C_6^{\nu'}\sigmaubseteq C_6^{(0,2)}$ for $\nu'\in\{(0,3),(0,4),(0,5),(0,6)\}$, because $\sigma_{Reg,1}=\rho_2=\{0\}\tauimes\mathbb R_{\gammaeq 0}$. By Proposition \rhoef{Lema2Gen}, only $\nu=(1,0)$ contributes to $F_6$ from the vectors in $H_6$, and the claim on $F_6$ follows. Note how at this level $\nu=(0,1)$ does no longer give rise to an irreducible component, since $l_2(0,1)=6$ and $(0,1)\notin N_2$. Then we have that $(0,2)\in F_6$ and the vertex associated with $C_5^{(0,1)}$ and the one associated with $C_6^{(0,2)}$ are joined in the graph $\Gamma'$.
\sigmaection{Technical results and proofs.}
\lambdaabel{Proofs}
In this section we state and prove some results which are used along the paper but only in the proofs of other results, and can be skipped to read Section \rhoef{Sec4}. Moreover there are some proofs which we leave to this section. | 4,042 | 114,405 | en |
train | 0.29.23 | \piut(6,70){$m=42$}\piut(25,70){$\betaullet$}\piut(45,70){$\betaullet$}\piut(80,70){$\betaullet$}
\piut(11,74){$\vdots$}\piut(25,74){$\vdots$}\piut(45,74){$\vdots$}\piut(80,74){$\vdots$}
\piut(6,80){$m=49$}\piut(25,80){$\betaullet$}\piut(45,80){$\betaullet$}\piut(80,80){$\betaullet$}
\piut(26,81){\lambdaine(0,1){5}}\piut(46,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(1,1){5}}
\piut(6,85){$m=50$}\piut(25,85){$\betaullet$}\piut(45,85){$\betaullet$}\piut(80,85){$\betaullet$}\piut(85,85){$\betaullet$}
\piut(26,86){\lambdaine(0,1){5}}\piut(46,86){\lambdaine(0,1){5}}\piut(81,86){\lambdaine(0,1){5}}\piut(86,86){\lambdaine(0,1){5}}
\piut(6,90){$m=51$}\piut(25,90){$\betaullet$}\piut(45,90){$\betaullet$}\piut(80,90){$\betaullet$}\piut(85,90){$\betaullet$}
\piut(26,91){\lambdaine(0,1){5}}\piut(26,91){\lambdaine(1,1){5}}\piut(46,91){\lambdaine(0,1){5}}\piut(81,91){\lambdaine(0,1){5}}\piut(86,91){\lambdaine(0,1){5}}
\piut(6,95){$m=52$}\piut(25,95){$\betaullet$}\piut(30,95){$\betaullet$}\piut(45,95){$\betaullet$}\piut(80,95){$\betaullet$}\piut(85,95){$\betaullet$}
\piut(26,96){\lambdaine(0,1){5}}\piut(31,96){\lambdaine(0,1){5}}\piut(46,96){\lambdaine(0,1){5}}\piut(81,96){\lambdaine(0,1){5}}\piut(86,97){$\nearrow$}
\piut(6,100){$m=53$}\piut(25,100){$\betaullet$}\piut(30,100){$\betaullet$}\piut(45,100){$\betaullet$}\piut(80,100){$\betaullet$}
\varepsilonnd{picture}
\varepsilonnd{center}
\caption{The graph of the surface defined by $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$.}
\lambdaabel{figEx1}
\varepsilonnd{figure}
The arrows in the figure represent the behaviour explained in Remark \rhoef{RemarkFinal}.
Now we give some explanations to illustrate how Proposition \rhoef{Lema2Gen} and Proposition \rhoef{PropC1} work.
For $m=1$, we have $H_1=\{(1,0),(0,1)\}$, $L_1^=\{(0,1)\}$ and $L_1^<=\varepsilonmptyset$. The claim on $F_1$ in this case follows easily by Proposition \rhoef{Lema2Gen}.
At level $m=6$ we have $H_6=\{\nu\in[0,m]^2\cap N_0\ |\ \nu_1\neq 0\}$, $L_6^==\varepsilonmptyset$ and $L_6^<=\{(0,\nu_2)\ |\ 2\lambdaeq\nu_2\lambdaeq 6\}$ with $j'(6,(0,2))=1$, then, by Proposition \rhoef{PropC1} $C_6^{\nu'}\sigmaubseteq C_6^{(0,2)}$ for $\nu'\in\{(0,3),(0,4),(0,5),(0,6)\}$, because $\sigma_{Reg,1}=\rho_2=\{0\}\tauimes\mathbb R_{\gammaeq 0}$. By Proposition \rhoef{Lema2Gen}, only $\nu=(1,0)$ contributes to $F_6$ from the vectors in $H_6$, and the claim on $F_6$ follows. Note how at this level $\nu=(0,1)$ does no longer give rise to an irreducible component, since $l_2(0,1)=6$ and $(0,1)\notin N_2$. Then we have that $(0,2)\in F_6$ and the vertex associated with $C_5^{(0,1)}$ and the one associated with $C_6^{(0,2)}$ are joined in the graph $\Gamma'$.
\sigmaection{Technical results and proofs.}
\lambdaabel{Proofs}
In this section we state and prove some results which are used along the paper but only in the proofs of other results, and can be skipped to read Section \rhoef{Sec4}. Moreover there are some proofs which we leave to this section.
Recall that we denote the first characteristic exponent by $\lambdaambda_1=\gammaamma_1=(\pihirac{a_1}{n_1},\pihirac{b_1}{n_1})$ with $a_1\gammaeq b_1,\ a_1>0\mbox{ and }b_1\gammaeq 0$,
and if $b_1=0$ then $a_1>n_1$ (we always consider normalized branches). If $g_1>0$ then, by Lemma \rhoef{Lema35}, we have that $n_{g_1+1}\gammaamma_{g_1+1}=
(\alphalpha_1^{(g_1+1)},\alphalpha_2^{(g_1+1)})+(r_1^{(g_1+1)}\gammaamma_1^{(1)}+\cdots+r_{g_1}^{(g_1+1)}\gammaamma_{g_1}^{(1)},0)$, therefore $n_{g_1+1}\gammaamma_{g_1+1}^{(2)}\in\mathbb Z$, or in other words, $\gammaamma_{g_1+1}=(\gammaamma_{g_1+1}^{(1)},\pihirac{b_{g_1+1}}{n_{g_1+1}})$
with $b_{g_1+1}\gammaeq 1$.
\betaegin{Lem}
In the relation $n_{g_2+1}\gammaamma_{g_2+1}=(\alphalpha_1^{(g_2+1)},\alphalpha_2^{(g_2+1)})+r_1^{(g_2+1)}\gammaamma_1+\cdots+r_{g_2}^{(g_2+1)}\gammaamma_{g_2}$ given in Lemma
\rhoef{LemaPedro}, we have that $\alphalpha_2^{(g_2+1)}>1$.
\lambdaabel{alphaM1}
\varepsilonnd{Lem}
{\varepsilonm Proof.} If $g_2=g_1$ the claim is trivial since $\alphalpha_2^{(g_2+1)}=b_{g_2+1}>1$. Otherwise $n_{g_2+1}\gammaamma_{g_2+1}^{(2)}=\alphalpha_2^{(g_2+1)}+
r_{g_2}^{(g_2+1)}\pihirac{1}{n_{g_2}}$, and since, by Lemma \rhoef{LemaPedro}, $\gammaamma_{g_2+1}^{(2)}\gammaeq n_{g_2}\gammaamma_{g_2}^{(2)}=1$ and $0\lambdaeq r_{g_2}^{(g_2+1)}<
n_{g_2}$, then $\alphalpha_2^{(g_2+1)}\gammaeq n_{g_2+1}-\pihirac{r_{g_2}^{(g_2+1)}}{n_{g_2}}>1$, because $n_{g_2+1}\gammaeq 2$. \pisifill $\Box$
In Corollary \rhoef{Corolario3} we describe the generators of $J_m^\nu$ for $\nu\in H_m\cup L_m$. But we also need to describe the polynomial $F_{\nu}^{(l_{i(\nu)}(\nu))}$ (recall that by definition $\nu\notin H_{l_{i(\nu)}(\nu)}\cup L_{l_{i(\nu)}(\nu)}$).
We do this in the next Lemma, but before we look at an example.
\betaegin{Exam}
Let $X$ be a quasi-ordinary surface defined by $f=((z^2-x_1^3x_2^2)^2-x_1^6x_2^4z)^3-x_1^{23}x_2^{14}z$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},1),\ \gammaamma_2=(\pihirac{15}{4},\pihirac{5}{2}) \mbox{ and }\gammaamma_3=(\pihirac{49}{6},5)$. Notice that $\nu=(0,3)\notin N_2$, and $l_2(\nu)=l_3(\nu)$. At level $m=45$ we have the set
\[D_{45}^{(0,3)}=V(x_2^{(0)},x_2^{(1)},x_2^{(2)},z^{(0)},z^{(1)},z^{(2)},F_{1,\nu}^{(6)},F_{1,\nu}^{(7)},F_{3,\nu}^{(45)})\cap D(x_1^{(0)})\cap D(x_2^{(3)}),\]
where
\[\betaegin{array}{rl}
F_{3,\nu}^{(45)} & ={F_{2,\nu}^{(15)}}^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
& =({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
= & ({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3(1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}),\\
\varepsilonnd{array}\]
since $D_{45}^{(0,3)}\sigmaubset D(z^{(3)})$. Since $\gammaamma(0)=(x_1^{(0)},0,0)\in X$, and we are considering germs of quasi-ordinary singularities, we have that $|x_1^{(0)}|<<1$ and we deduce that $1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}\neq 0$.
\lambdaabel{Ex4}
\varepsilonnd{Exam}
This example illustrates the fact that we are looking at jet schemes of a germ of quasi-ordinary singularity, instead of jet schemes of the whole affine surface. If we looked at the whole surface there would be other irreducible components that we do not consider here. This is expectable because the components we consider are determined by the invariants of the topological type at the origin, so they describe only what happens in a small neighbourhood of zero. Actually the other components that may appear when looking at the whole affine surface, will project on closed points, different from the origin, of the singular locus.
\betaegin{Lem}
Given $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ with $m+1=l_{i(\nu)}(\nu)$, then
\[F_\nu^{(l_{i(\nu)}(\nu))}= \lambdaeft({x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\rhoight)^{e_{i(\nu)}}\cdot U,\]
where $U$ is a unit in $\mathbb C[[{x_1^{(\nu_1)}}^{\pim 1},{x_2^{(\nu_2)}}^{\pim 1},{F_0^{(\pihirac{l_1(\nu)}{e_0})}}^{\pim 1},\lambdadots,{F_{i(\nu)-2}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{\pim 1}]]$.
When $\nu\notin\rho_1\cup\rho_2$, then $U=1$.
\lambdaabel{LemF'}
\varepsilonnd{Lem} | 3,446 | 114,405 | en |
train | 0.29.24 | {\varepsilonm Proof.} If $g_2=g_1$ the claim is trivial since $\alphalpha_2^{(g_2+1)}=b_{g_2+1}>1$. Otherwise $n_{g_2+1}\gammaamma_{g_2+1}^{(2)}=\alphalpha_2^{(g_2+1)}+
r_{g_2}^{(g_2+1)}\pihirac{1}{n_{g_2}}$, and since, by Lemma \rhoef{LemaPedro}, $\gammaamma_{g_2+1}^{(2)}\gammaeq n_{g_2}\gammaamma_{g_2}^{(2)}=1$ and $0\lambdaeq r_{g_2}^{(g_2+1)}<
n_{g_2}$, then $\alphalpha_2^{(g_2+1)}\gammaeq n_{g_2+1}-\pihirac{r_{g_2}^{(g_2+1)}}{n_{g_2}}>1$, because $n_{g_2+1}\gammaeq 2$. \pisifill $\Box$
In Corollary \rhoef{Corolario3} we describe the generators of $J_m^\nu$ for $\nu\in H_m\cup L_m$. But we also need to describe the polynomial $F_{\nu}^{(l_{i(\nu)}(\nu))}$ (recall that by definition $\nu\notin H_{l_{i(\nu)}(\nu)}\cup L_{l_{i(\nu)}(\nu)}$).
We do this in the next Lemma, but before we look at an example.
\betaegin{Exam}
Let $X$ be a quasi-ordinary surface defined by $f=((z^2-x_1^3x_2^2)^2-x_1^6x_2^4z)^3-x_1^{23}x_2^{14}z$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},1),\ \gammaamma_2=(\pihirac{15}{4},\pihirac{5}{2}) \mbox{ and }\gammaamma_3=(\pihirac{49}{6},5)$. Notice that $\nu=(0,3)\notin N_2$, and $l_2(\nu)=l_3(\nu)$. At level $m=45$ we have the set
\[D_{45}^{(0,3)}=V(x_2^{(0)},x_2^{(1)},x_2^{(2)},z^{(0)},z^{(1)},z^{(2)},F_{1,\nu}^{(6)},F_{1,\nu}^{(7)},F_{3,\nu}^{(45)})\cap D(x_1^{(0)})\cap D(x_2^{(3)}),\]
where
\[\betaegin{array}{rl}
F_{3,\nu}^{(45)} & ={F_{2,\nu}^{(15)}}^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
& =({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
= & ({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3(1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}),\\
\varepsilonnd{array}\]
since $D_{45}^{(0,3)}\sigmaubset D(z^{(3)})$. Since $\gammaamma(0)=(x_1^{(0)},0,0)\in X$, and we are considering germs of quasi-ordinary singularities, we have that $|x_1^{(0)}|<<1$ and we deduce that $1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}\neq 0$.
\lambdaabel{Ex4}
\varepsilonnd{Exam}
This example illustrates the fact that we are looking at jet schemes of a germ of quasi-ordinary singularity, instead of jet schemes of the whole affine surface. If we looked at the whole surface there would be other irreducible components that we do not consider here. This is expectable because the components we consider are determined by the invariants of the topological type at the origin, so they describe only what happens in a small neighbourhood of zero. Actually the other components that may appear when looking at the whole affine surface, will project on closed points, different from the origin, of the singular locus.
\betaegin{Lem}
Given $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ with $m+1=l_{i(\nu)}(\nu)$, then
\[F_\nu^{(l_{i(\nu)}(\nu))}= \lambdaeft({x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\rhoight)^{e_{i(\nu)}}\cdot U,\]
where $U$ is a unit in $\mathbb C[[{x_1^{(\nu_1)}}^{\pim 1},{x_2^{(\nu_2)}}^{\pim 1},{F_0^{(\pihirac{l_1(\nu)}{e_0})}}^{\pim 1},\lambdadots,{F_{i(\nu)-2}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{\pim 1}]]$.
When $\nu\notin\rho_1\cup\rho_2$, then $U=1$.
\lambdaabel{LemF'}
\varepsilonnd{Lem}
{\varepsilonm Proof.} We have that $j(m,\nu)=i(\nu)-1$, and, by Lemma \rhoef{Lemfk}, for any $\gammaamma\in D_m^\nu$
\[\betaegin{array}{ll}
\mbox{ord}_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle & \mbox{ for }0\lambdaeq k\lambdaeq i(\nu)-2\\
\\
\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)>\pihirac{m}{e_{i(\nu)-1}}\\
\varepsilonnd{array}\]
Then ord$_t(f_{i(\nu)-1}\circ\gammaamma)\gammaeq\pihirac{m+1}{e_{i(\nu)-1}}=\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)-1}}=\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, and since $\nu\notin N_{i(\nu)}$, $\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$ is not an integer. Hence
\[\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)>\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle.\]
We have by Lemma \rhoef{Lema35}
\[f_{i(\nu)}=f_{i(\nu)-1}^{n_{i(\nu)}}-c_{i(\nu)}x_1^{\alphalpha_1^{(i(\nu))}}x_2^{\alphalpha_2^{(i(\nu))}}f_0^{r_1^{(i(\nu))}}\cdots f_{i(\nu)-2}^{r_{i(\nu)-1}^{(i(\nu))}}+\sigmaum c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}f_0^{r_1}\cdots f_{i(\nu)-1}^{r_{i(\nu)}},\]
and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{i(\nu)-1}^{n_{i(\nu)}}\circ\gammaamma) & >n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle\\
\\
\mbox{ord}_t((c_{i(\nu)}x_1^{\alphalpha_1^{(i(\nu))}}x_2^{\alphalpha_2^{(i(\nu))}}\cdots f_{i(\nu)-2}^{r_{i(\nu)-1}^{(i(\nu))}})\circ\gammaamma) & =n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle\\
\\
\mbox{ord}_t((c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}\cdots f_{i(\nu)-1}^{r_{i(\nu)}})\circ\gammaamma) &
=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i(\nu)-1}\gammaamma_{i(\nu)-1}\rhoangle+r_{i(\nu)}\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)\\
& >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i(\nu)}\gammaamma_{i(\nu)}\rhoangle\\
& \gammaeq n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle.\\
\varepsilonnd{array}\]
Then ord$_t(f_{i(\nu)}\circ\gammaamma)=n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle=\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}}$, and
\[F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}= {x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\]
By Lemma \rhoef{LemExpSR}
\[f=f_{i(\nu)}^{e_{i(\nu)}}+\sigmaum c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k\]
where $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$. Then
\[\betaegin{array}{ll}
\mbox{ord}_t(f_{i(\nu)}^{e_{i(\nu)}}\circ\gammaamma)=l_{i(\nu)}(\nu)\\
\\
\mbox{ord}_t((c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k)\circ\gammaamma)\gammaeq l_{i(\nu)}(\nu)\\
\varepsilonnd{array}\]
and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}+G_{i(\nu),\nu},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
and the sum runs over $i,j,k$ such that
\betaegin{enumerate}
\item[(i)] $c_{ijk}^{(i(\nu))}\neq 0$
\item[(ii)] $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=l_{i(\nu)}(\nu)$
\varepsilonnd{enumerate}
Notice that if $\nu\notin\rho_1\cup\rho_2$, then condition (ii) never holds and $G_{\nu,i(\nu)}=0$. In this case we are done. Otherwise, from (\rhoef{eqO}) we deduce that $\gammaamma\in D_m^\nu\sigmaubset D(F_{\nu,i(\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})})$, and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\lambdaeft(1+G_{i(\nu),\nu}/{F_{i(\nu,\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}
^{e_{i(\nu)}}\rhoight).\]
\pisifill$\Box$
{\betaf\varepsilonm Proof of Lemma \rhoef{LemD}.}
If $m<l_{i(\nu)}(\nu)$, then we deduce from Proposition \rhoef{Prop1} that $D_m^\nu$ is non-empty.
Otherwise $m\gammaeq l_{i(\nu)}(\nu)$, and by definition $F^{(l_{i(\nu)}(\nu))}\in J_m^\nu$, and by Lemma \rhoef{LemF'} (and its proof)
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\cdot U,\]
where
\[F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}={x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\]
with $(\alphalpha_1^{(i(\nu))},\alphalpha_2^{(i(\nu))})+r_1^{(i(\nu))}\gammaamma_1+\cdots+r_{i(\nu)-1}^{(i(\nu))}\gammaamma_{i(\nu)-1}=n_{i(\nu)}\gammaamma_{i(\nu)}$. And
\[U=1+G_{i(\nu),\nu}/{F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
with $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=n_{i(\nu)}e_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, though $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$.
Notice that at level $l_{i(\nu)}(\nu)-1$ we can apply Corollary \rhoef{Corolario3}, and deduce, as in (\rhoef{eqO}), that
\[D_{l_{i(\nu)-1}(\nu)-1}^\nu\sigmaubseteq D(F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}\cdots F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})})\]
Hence $D_{l_{i(\nu)}(\nu)}^\nu$ satisfies the same property, and since, by definition, $D_{l_{i(\nu)}(\nu)}^\nu\sigmaubseteq D(x_1^{(\nu_1)}\cdot x_2^{(\nu_2)})$, we just have to argue that $U\neq 0$. | 4,073 | 114,405 | en |
train | 0.29.25 | By Lemma \rhoef{LemExpSR}
\[f=f_{i(\nu)}^{e_{i(\nu)}}+\sigmaum c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k\]
where $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$. Then
\[\betaegin{array}{ll}
\mbox{ord}_t(f_{i(\nu)}^{e_{i(\nu)}}\circ\gammaamma)=l_{i(\nu)}(\nu)\\
\\
\mbox{ord}_t((c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k)\circ\gammaamma)\gammaeq l_{i(\nu)}(\nu)\\
\varepsilonnd{array}\]
and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}+G_{i(\nu),\nu},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
and the sum runs over $i,j,k$ such that
\betaegin{enumerate}
\item[(i)] $c_{ijk}^{(i(\nu))}\neq 0$
\item[(ii)] $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=l_{i(\nu)}(\nu)$
\varepsilonnd{enumerate}
Notice that if $\nu\notin\rho_1\cup\rho_2$, then condition (ii) never holds and $G_{\nu,i(\nu)}=0$. In this case we are done. Otherwise, from (\rhoef{eqO}) we deduce that $\gammaamma\in D_m^\nu\sigmaubset D(F_{\nu,i(\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})})$, and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\lambdaeft(1+G_{i(\nu),\nu}/{F_{i(\nu,\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}
^{e_{i(\nu)}}\rhoight).\]
\pisifill$\Box$
{\betaf\varepsilonm Proof of Lemma \rhoef{LemD}.}
If $m<l_{i(\nu)}(\nu)$, then we deduce from Proposition \rhoef{Prop1} that $D_m^\nu$ is non-empty.
Otherwise $m\gammaeq l_{i(\nu)}(\nu)$, and by definition $F^{(l_{i(\nu)}(\nu))}\in J_m^\nu$, and by Lemma \rhoef{LemF'} (and its proof)
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\cdot U,\]
where
\[F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}={x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\]
with $(\alphalpha_1^{(i(\nu))},\alphalpha_2^{(i(\nu))})+r_1^{(i(\nu))}\gammaamma_1+\cdots+r_{i(\nu)-1}^{(i(\nu))}\gammaamma_{i(\nu)-1}=n_{i(\nu)}\gammaamma_{i(\nu)}$. And
\[U=1+G_{i(\nu),\nu}/{F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
with $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=n_{i(\nu)}e_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, though $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$.
Notice that at level $l_{i(\nu)}(\nu)-1$ we can apply Corollary \rhoef{Corolario3}, and deduce, as in (\rhoef{eqO}), that
\[D_{l_{i(\nu)-1}(\nu)-1}^\nu\sigmaubseteq D(F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}\cdots F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})})\]
Hence $D_{l_{i(\nu)}(\nu)}^\nu$ satisfies the same property, and since, by definition, $D_{l_{i(\nu)}(\nu)}^\nu\sigmaubseteq D(x_1^{(\nu_1)}\cdot x_2^{(\nu_2)})$, we just have to argue that $U\neq 0$.
In Lemma \rhoef{LemF'} we prove that $U=1$ when $\nu\notin\rho_1\cup\rho_2$. Suppose the contrary, then either $\nu_1=0$ or $\nu_2=0$. We work it all out and write $F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}$ in terms of $x_1^{(\nu_1)}$, $x_2^{(\nu_2)}$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$. The key point is that when $G_{i(\nu),\nu}\neq 0$ is because $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$ but $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=n_{i(\nu)}e_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, because $\nu\in\rho_1\cup\rho_2$. Therefore either $x_1^{(0)}$ or $x_2^{(0)}$ do appear in U (it do not cancel in the quotient $G_{i(\nu),\nu}/{F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}$), depending on whether $\nu_1=0$ or $\nu_2=0$. Then we have that $U$ depends on the origin of the jet $\gammaamma(0)\in X$, and since we are dealing with germs of quasi-ordinary singularities, we have that $|x_i^{(0)}|<<1$ for $i=1,2$, and we deduce that $U\neq 0$.
\pisifill$\Box$
{\betaf\varepsilonm Proof of Lemma \rhoef{Lemfk}.} We distinguish the cases $\nu\in H_m$ and $\nu\in L_m$. For $\nu\in H_m$, $j(m,\nu)=0$ and we have to prove that $\mbox{ord}_t(f_k\circ\gammaamma)>\pihirac{m}{e_k}$ for $0\lambdaeq k\lambdaeq g$. By Proposition \rhoef{CHm} it is true for $k=0$. For $k=g$ the claim is obvious, and for $1\lambdaeq k\lambdaeq g-1$ we use the expansion (\rhoef{eqSRpqprima}) in Lemma \rhoef{LemExpSR},
\[f=f_k^{e_k}+\sigmaum_{(i,j)+r\gammaamma_1>n_ke_k\gammaamma_k}c_{ijr}^{(k)}x_1^ix_2^jz^r.\]
Suppose that there exist $c_{ijr}^{(k)}\neq 0$ such that $\lambdaangle\nu,(i,j)\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq m$. Then, using that ord$_t(z\circ\gammaamma)>\pihirac{m}{n}$ and $k\gammaeq 1$, we have the following inequalities
\[l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\pihirac{m}{n}<l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq l_k(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq m.\]
Then $l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle<m(1-\pihirac{r}{n})$, and since $r<n$ this is equivalent to $\lambdaangle\nu,\gammaamma_1\rhoangle<\pihirac{m}{n}$, which contradicts the fact that $j(m,\nu)=0$. Hence ord$_t(c_{ijr}^{(k)}x_1^ix_2^jz^r\circ\gammaamma)>m$ and therefore ord$_t(f_k^{e_k}\circ\gammaamma)>m$ as we wanted to prove.
For $\nu\in L_m$ the proof is by induction on $j(m,\nu)$. And we will make use repeatedly of the following observation. In general, for any function $f\in\mathbb C[x_1,x_2,z]$, and any $m$-jet $\gammaamma$, there is no relation among ord$_t(f\circ\gammaamma)$ and ord$_t(f\circ\pii_{m,m'}(\gammaamma))$ with $m'<m$. But if ord$_t(x_i\circ\gammaamma)\neq 0$ for $i=1,2$, ord$_t(z\circ\gammaamma)\neq 0$, and for $m'<m$, the $m'$-jet $\gammaamma':=\pii_{m,m'}(\gammaamma)$ verifies that ord$_t(x_i\circ\gammaamma')\neq 0$ and ord$_t(z\circ\gammaamma')\neq 0$, then
\[\mbox{ord}_t(f\circ\gammaamma)=\mbox{ord}_t(f\circ\gammaamma').\]
By Remark \rhoef{RemDefHL} (ii) the first case of induction is $j(m,\nu)=1$. Then in particular $l_1(\nu)<l_2(\nu)$. For $\gammaamma\in D_m^\nu$, set $\betaar m:=l_1(\nu)-1<m$ and $\betaar\gammaamma:=\pii_{m,\betaar m}(\gammaamma)$. Then $j(\betaar m,\nu)=0$ and $\betaar\gammaamma\in D_{\betaar m}^\nu$. By Proposition \rhoef{CHm} we have
\[\mbox{ord}_t(z\circ\betaar\gammaamma)>\pihirac{\betaar m}{n}=\pihirac{l_1(\nu)-1}{n}=\lambdaangle\nu,\gammaamma_1\rhoangle-\pihirac{1}{n}\]
Hence ord$_t(z\circ\betaar\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$ and therefore ord$_t(z\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$. Suppose that the inequality is strict, ord$_t(z\circ\gammaamma)>\lambdaangle\nu,\gammaamma_1\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_1=z^{n_1}-x_1^{a_1}x_2^{b_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1\gammaamma_1}x_1^{i_1}x_2^{i_2}z^k\]
Then
\[\betaegin{array}{rl}
\mbox{ord}_t(z^{n_1}\circ\gammaamma) & >n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{a_1}x_2^{b_1}\circ\gammaamma) & =n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\varepsilonnd{array}\]
and hence ord$_t(f_1\circ\gammaamma)=n_1\lambdaangle\nu,\gammaamma_1\rhoangle$. Again by Lemma \rhoef{LemExpSR} we have
\[f=f_1^{e_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1e_1\gammaamma_1}c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\]
and since
\[\betaegin{array}{rl}
\mbox{ord}_t(f_1^{e_1}\circ\gammaamma) & =e_1n_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\varepsilonnd{array}\]
we deduce ord$_t(f\circ\gammaamma)=l_1(\nu)\lambdaeq m$, which is a contradiction. Then ord$_t(z\circ\gammaamma)=\lambdaangle\nu,\gammaamma_1\rhoangle$. | 3,670 | 114,405 | en |
train | 0.29.26 | For $\nu\in L_m$ the proof is by induction on $j(m,\nu)$. And we will make use repeatedly of the following observation. In general, for any function $f\in\mathbb C[x_1,x_2,z]$, and any $m$-jet $\gammaamma$, there is no relation among ord$_t(f\circ\gammaamma)$ and ord$_t(f\circ\pii_{m,m'}(\gammaamma))$ with $m'<m$. But if ord$_t(x_i\circ\gammaamma)\neq 0$ for $i=1,2$, ord$_t(z\circ\gammaamma)\neq 0$, and for $m'<m$, the $m'$-jet $\gammaamma':=\pii_{m,m'}(\gammaamma)$ verifies that ord$_t(x_i\circ\gammaamma')\neq 0$ and ord$_t(z\circ\gammaamma')\neq 0$, then
\[\mbox{ord}_t(f\circ\gammaamma)=\mbox{ord}_t(f\circ\gammaamma').\]
By Remark \rhoef{RemDefHL} (ii) the first case of induction is $j(m,\nu)=1$. Then in particular $l_1(\nu)<l_2(\nu)$. For $\gammaamma\in D_m^\nu$, set $\betaar m:=l_1(\nu)-1<m$ and $\betaar\gammaamma:=\pii_{m,\betaar m}(\gammaamma)$. Then $j(\betaar m,\nu)=0$ and $\betaar\gammaamma\in D_{\betaar m}^\nu$. By Proposition \rhoef{CHm} we have
\[\mbox{ord}_t(z\circ\betaar\gammaamma)>\pihirac{\betaar m}{n}=\pihirac{l_1(\nu)-1}{n}=\lambdaangle\nu,\gammaamma_1\rhoangle-\pihirac{1}{n}\]
Hence ord$_t(z\circ\betaar\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$ and therefore ord$_t(z\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$. Suppose that the inequality is strict, ord$_t(z\circ\gammaamma)>\lambdaangle\nu,\gammaamma_1\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_1=z^{n_1}-x_1^{a_1}x_2^{b_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1\gammaamma_1}x_1^{i_1}x_2^{i_2}z^k\]
Then
\[\betaegin{array}{rl}
\mbox{ord}_t(z^{n_1}\circ\gammaamma) & >n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{a_1}x_2^{b_1}\circ\gammaamma) & =n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\varepsilonnd{array}\]
and hence ord$_t(f_1\circ\gammaamma)=n_1\lambdaangle\nu,\gammaamma_1\rhoangle$. Again by Lemma \rhoef{LemExpSR} we have
\[f=f_1^{e_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1e_1\gammaamma_1}c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\]
and since
\[\betaegin{array}{rl}
\mbox{ord}_t(f_1^{e_1}\circ\gammaamma) & =e_1n_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\varepsilonnd{array}\]
we deduce ord$_t(f\circ\gammaamma)=l_1(\nu)\lambdaeq m$, which is a contradiction. Then ord$_t(z\circ\gammaamma)=\lambdaangle\nu,\gammaamma_1\rhoangle$.
Now we prove that ord$_t(f_1\circ\gammaamma)>\pihirac{m}{e_1}$. Suppose the contrary, ord$_t(f_1\circ\gammaamma)\lambdaeq\pihirac{m}{e_1}$. If $g=1$ there is nothing to prove.
If $g=2$, then we consider the expansion given in Lemma \rhoef{Lema35}
\[f=f_2=f_1^{n_2}-c_2x_1^{\alphalpha_1^{(2)}}x_2^{\alphalpha_2^{(2)}}z^{r_1^{(2)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{i_1}x_2^{i_2}z^{r_1}f_1^{r_2}\]
where $(i_1,i_2)+r_1\gammaamma_1+r_2\gammaamma_2>n_2\gammaamma_2$, and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_1^{n_2}\circ\gammaamma) & \lambdaeq m,\\
\\
\mbox{ord}_t(x_1^{\alphalpha_1^{(2)}}x_2^{\alphalpha_2^{(2)}}z^{r_1^{(2)}}\circ\gammaamma) & =n_2\lambdaangle\nu,\gammaamma_2\rhoangle=l_2(\nu)>m\\
\varepsilonnd{array}\]
Then there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
\[n_2\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r_1\gammaamma_1\rhoangle+r_2\mbox{ord}_t(f_1\circ\gammaamma)\]
or equivalently
\[(n_2-r_2)\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r_1\gammaamma_1\rhoangle\gammaeq (n_2-r_2)\lambdaangle\nu,\gammaamma_2\rhoangle\]
And since $r_2<n_2$ we conclude ord$_t(f_1\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_2\rhoangle>\pihirac{m}{e_1}$, which is a contradiction.
If $g>2$, by Lemma \rhoef{LemExpSR}
\[f=f_1^{e_1}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}f_1^{r_2}\]
and since we are supposing that ord$_t(f_1^{e_1}\circ\gammaamma)\lambdaeq m$, there must exists $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
$\mbox{ord}_t(f_1^{e_1}\circ\gammaamma)=\mbox{ord}_t(x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}f_1^{r_2}\circ\gammaamma)$,
hence
\[e_1\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1\rhoangle+r_2\mbox{ord}_t(f_1\circ\gammaamma)\]
or equivalently
\[(e_1-r_2)\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1\rhoangle\gammaeq n_2e_2\lambdaangle\nu,\gammaamma_2\rhoangle-r_2\lambdaangle\nu,\gammaamma_2\rhoangle\]
and since $r_2<e_1$ we conclude that ord$_t(f_1\circ\gammaamma)\gammaeq \lambdaangle\nu,\gammaamma_2\rhoangle$, which is a contradiction.
The rest is simple, by Lemma \rhoef{LemExpSR}, for $k>1=j(m,\nu)$, we have
\[f=f_k^{e_k}+\sigmaum c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\]
and since ord$_t(c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq l_k(\nu)>m$, we deduce ord$_t(f_k^{e_k}\circ\gammaamma)>m$.
Suppose now that the claim is true for $j(m,\nu)=j$ and we will prove it for $j(m,\nu)=j+1$. Let $\gammaamma\in D_m^\nu$, with
$l_{j+1}(\nu)\lambdaeq m<l_{j+1}(\nu)$. We set $\betaar m=l_{j+1}(\nu)-1$ and $\betaar\gammaamma=\pii_{m,\betaar m}(\gammaamma)$. Then $\gammaamma\in D_{\betaar m}^\nu$ and $j(\betaar m,\nu)=i\lambdaeq j$, where
\[l_i(\nu)\lambdaeq\betaar m<l_{i+1}(\nu)=\cdots=l_{j+1}(\nu).\]
Then, by Lemma \rhoef{l-ord}, this is equivalent to
\[\betaegin{array}{c}
n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle=\lambdaangle\nu,\gammaamma_{i+2}\rhoangle\\
\\
n_{i+2}\lambdaangle\nu,\gammaamma_{i+2}\rhoangle=\lambdaangle\nu,\gammaamma_{i+3}\rhoangle\\
\vdots\\
n_j\lambdaangle\nu,\gammaamma_j\rhoangle=\lambdaangle\nu,\gammaamma_{j+1}\rhoangle\\
\varepsilonnd{array}\]
By induction hypothesis we deduce ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k<i$. We are going to prove that ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By induction we have
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\mbox{ord}_t(f_i\circ\betaar\gammaamma)>\pihirac{\betaar m}{e_i}=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle-\pihirac{1}{e_i}\]
Therefore
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\]
Let us suppose that ord$_t(f_i\circ\gammaamma)>\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_{i+1}=f_i^{n_{i+1}}-c_{i+1}x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\cdots f_{i-1}^{r_i^{(i+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
and we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_i^{n_{i+1}}\circ\gammaamma) & > n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\circ\gammaamma) & =n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\varepsilonnd{array}\]
Therefore ord$_t(f_{i+1}\circ\gammaamma)=n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f=f_{i+1}^{e_{i+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$. Then
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{i+1}^{e_{i+1}}\circ\gammaamma) & =l_{i+1}(\nu)\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Then ord$_t(f\circ\gammaamma)=l_{i+1}(\nu)=l_{j+1}(\nu)\lambdaeq m$, which is a contradiction. Therefore ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We can prove that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $i<k\lambdaeq j$ one after the other exactly as the proof of ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We prove now that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$. Suppose the contrary, ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$. | 3,870 | 114,405 | en |
train | 0.29.27 | By induction hypothesis we deduce ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k<i$. We are going to prove that ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By induction we have
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\mbox{ord}_t(f_i\circ\betaar\gammaamma)>\pihirac{\betaar m}{e_i}=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle-\pihirac{1}{e_i}\]
Therefore
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\]
Let us suppose that ord$_t(f_i\circ\gammaamma)>\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_{i+1}=f_i^{n_{i+1}}-c_{i+1}x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\cdots f_{i-1}^{r_i^{(i+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
and we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_i^{n_{i+1}}\circ\gammaamma) & > n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\circ\gammaamma) & =n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\varepsilonnd{array}\]
Therefore ord$_t(f_{i+1}\circ\gammaamma)=n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f=f_{i+1}^{e_{i+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$. Then
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{i+1}^{e_{i+1}}\circ\gammaamma) & =l_{i+1}(\nu)\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Then ord$_t(f\circ\gammaamma)=l_{i+1}(\nu)=l_{j+1}(\nu)\lambdaeq m$, which is a contradiction. Therefore ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We can prove that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $i<k\lambdaeq j$ one after the other exactly as the proof of ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We prove now that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$. Suppose the contrary, ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$.
If $j+2=g$ then by Lemma \rhoef{LemExpSR}
\[f=f_{j+2}=f_{j+1}^{n_{j+2}}-c_{j+2}x_1^{\alphalpha_1^{(g)}}x_2^{(g)}z^{r_1^{(g)}}\cdots f_j^{r_{j+1}^{(g)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}} x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
while, if $j+2<g$ we have the expansion
\[f=f_{j+1}^{e_{j+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}} x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
with $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+2}\gammaamma_{j+2}>n_{j+2}e_{j+2}\gammaamma_{j+2}$. In both cases we have
\[f=f_{j+1}^{e_{j+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
with $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+2}\gammaamma_{j+2}\gammaeq n_{j+2}e_{j+2}\gammaamma_{j+2}$. Looking at the expansion, since ord$_t(f_{j+1}^{e_{j+1}}\circ\gammaamma)\lambdaeq m$, there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
ord$_t(f_{j+1}^{e_{j+1}}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}\rhoangle+r_{j+2}\mbox{ord}_t(f_{j+1}\circ\gammaamma)$, or equivalently
\[(e_{j+1}-r_{j+2})\mbox{ord}_t(f_{j+1}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}\rhoangle\gammaeq n_{j+2}e_{j+2}\lambdaangle\nu,\gammaamma_{j+2}\rhoangle-r_{j+2}\lambdaangle\nu,\gammaamma_{j+2}\rhoangle\]
And since $r_{j+2}<e_{j+1}$ we deduce that ord$_t(f_{j+1}\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_{j+2}\rhoangle>\pihirac{m}{e_{j+1}}$, which is a contradiction.
Finally we prove that ord$_t(f_k^{e_k}\circ\gammaamma)>m$. By Lemma \rhoef{LemExpSR}, $f=f_k^{e_k}+\sigmaum c_{i_1i_2r}x_1^{i_1}x_2^{i_2}z^r$
with $(i_1,i_2)+r\gammaamma_1>n_ke_k\gammaamma_k$. Then ord$_t(x_1^{i_1}x_2^{i_2}z^r\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r\gammaamma_1\rhoangle\gammaeq l_k(\nu)>m$, and the result follows.
\pisifill$\Box$
\betaegin{Lem} For $m\in\mathbb Z_{>0}$ and $\nu\in L_m$, we have the following.
\betaegin{enumerate}
\item[(i)] If $i\lambdaeq j(m,\nu)$, then
\[F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}=0\mbox{ if and only if }{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}=0\]
Roughly speaking the part $G_{i,\nu}$ in equation (\rhoef{EqP}) is not meaningful.
\
\item[(ii)] For $m(\nu)\lambdaeq j\lambdaeq g_1$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})\cap
D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}),\]
and for $g_1<j\lambdaeq g$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle)})\cap D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}).\]
\varepsilonnd{enumerate}
\lambdaabel{TechLem}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
(i) First observe that if $j(m,\nu)\gammaeq i$, then $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}\in J_m^\nu$ and it has the form given in equation (\rhoef{EqP}) in Corollary \rhoef{Corolario3}. The proof is obvious if $G_{i,\nu}=0$. Suppose the contrary. Then $\nu\in\rho_1\cup\rho_2$, and the claim is not obvious if $\betaar G_{i,\nu}\neq 0$, where
\[\betaar G_{i,\nu}=\sigmaum c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}\]
with the conditions $c_{\underline{\alphalpha},\underline{r}}\neq 0$, $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$ and $r_i=0$. Then, $F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}=0$ if and only if
\[c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+\betaar G_{i,\nu}=0\]
where, remember that any term $c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}$ in particular appears in $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}$, and hence it satisfies that $(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i-1}\gammaamma_{i-1}$. Then we can write last equation as
\[{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\lambdaeft(c_i+P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,F_{i-2,\nu}^
{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})})\rhoight)\]
where $P$ is a polynomial non-unit. If we work it all out, then we can write $P$ as a polynomial in $x_1^{(\nu_1)}$, $x_2^{(\nu_2)}$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$. Now we use that $\nu\in\rho_1\cup\rho_2$, and hence either $\nu_1=0$ or $\nu_2=0$. Then $P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})$ depends on the origin $\gammaamma(0)$ of the jets, and, since we are dealing with germs of quasi-ordinary singularities we can always suppose that $|x_i^{(\nu_i)}|<<1$ for $i=1$ or $2$ depending on whether $\nu_1=0$ or $\nu_2=0$. As a consequence we can always suppose that $P<<c_i$. | 3,895 | 114,405 | en |
train | 0.29.28 | \
\item[(ii)] For $m(\nu)\lambdaeq j\lambdaeq g_1$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})\cap
D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}),\]
and for $g_1<j\lambdaeq g$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle)})\cap D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}).\]
\varepsilonnd{enumerate}
\lambdaabel{TechLem}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
(i) First observe that if $j(m,\nu)\gammaeq i$, then $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}\in J_m^\nu$ and it has the form given in equation (\rhoef{EqP}) in Corollary \rhoef{Corolario3}. The proof is obvious if $G_{i,\nu}=0$. Suppose the contrary. Then $\nu\in\rho_1\cup\rho_2$, and the claim is not obvious if $\betaar G_{i,\nu}\neq 0$, where
\[\betaar G_{i,\nu}=\sigmaum c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}\]
with the conditions $c_{\underline{\alphalpha},\underline{r}}\neq 0$, $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$ and $r_i=0$. Then, $F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}=0$ if and only if
\[c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+\betaar G_{i,\nu}=0\]
where, remember that any term $c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}$ in particular appears in $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}$, and hence it satisfies that $(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i-1}\gammaamma_{i-1}$. Then we can write last equation as
\[{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\lambdaeft(c_i+P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,F_{i-2,\nu}^
{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})})\rhoight)\]
where $P$ is a polynomial non-unit. If we work it all out, then we can write $P$ as a polynomial in $x_1^{(\nu_1)}$, $x_2^{(\nu_2)}$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$. Now we use that $\nu\in\rho_1\cup\rho_2$, and hence either $\nu_1=0$ or $\nu_2=0$. Then $P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})$ depends on the origin $\gammaamma(0)$ of the jets, and, since we are dealing with germs of quasi-ordinary singularities we can always suppose that $|x_i^{(\nu_i)}|<<1$ for $i=1$ or $2$ depending on whether $\nu_1=0$ or $\nu_2=0$. As a consequence we can always suppose that $P<<c_i$.
(ii) The inclusions follow directly from Corollary \rhoef{Corolario3} when $G_{j,\nu}=0$. If $G_{j,\nu}\neq 0$, the proof is by induction on $j$. For $j=m(\nu)$ the claim says that
\[V(I^\nu,F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})})\cap D(M)\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}),\]
where
\[M=\lambdaeft\{\betaegin{array}{cl}
x_1^{(\nu_1)} & \mbox{ if }g_1\gammaeq 1\\
\\
x_1^{(\nu_1)}x_2^{(\nu_2)} & \mbox{ if }g_1=0\\
\varepsilonnd{array}\rhoight.\]
and it follows by Lemma \rhoef{Lemfk}. Suppose it is true for $j-1$ and we prove it for $j$. We only have to prove that $F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}\neq 0$. But this follows by (i).
\pisifill$\Box$
\betaibliographystyle{amsplain}
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train | 0.30.0 | \begin{document}
\title[Nakamaye's theorem on log canonical pairs]{Nakamaye's theorem on log canonical pairs}
\dedicatory{\normalsize \dag \ Dedicated to the memory of Fassi}
\author[S. Cacciola and A.F. Lopez]{Salvatore Cacciola* and Angelo Felice Lopez*}
\thanks{* Research partially supported by the MIUR national project ``Geometria delle variet\`a
algebriche" PRIN 2010-2011.}
\address{\hspace{.1in}kip -.43cm Dipartimento di Matematica e Fisica, Universit\`a di Roma
Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. e-mail {\tt [email protected], [email protected]}}
\thanks{{^{-1}t 2010 Mathematics Subject Classification} : Primary 14C20, 14F18. Secondary 14E15, 14B05}
\begin{abstract}
We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension $\leq 1$. We also generalize Ein-Lazarsfeld-Musta{\c{t}}{\u{a}}-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.
\end{abstract}
\maketitle
\section{Introduction}
\label{intro}
Let $X$ be a normal complex projective variety and let $D$ be a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. The stable base locus
\[ \mathbf{B}(D)= \bigcap_{E \geq 0 : E \sim _{{\mathbb Q}} D} {\rm Supp}(E) \]
is an important closed subset associated to $D$, but it is often difficult to handle. On the other hand, there are two, perhaps even more important, base loci associated to $D$.
One of them is the augmented base locus (\cite{n}, \cite[Def. 1.2]{elmnp1})
\[ \mathbf{B}_+(D)= \bigcap_{E \geq 0 : D - E \mbox{\ ample}} {\rm Supp}(E) \]
where $E$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor. Since this locus measures the failure of $D$ to be ample, it has proved to be a key tool in several recent important results
in birational geometry, such as Takayama \cite{t}, Hacon and McKernan's \cite{hm} effective birationality of pluricanonical maps or Birkar, Cascini, Hacon and McKernan's \cite{bchm} finite generation of the canonical ring, just to mention a few.
One way to compute $\mathbf{B}_+(D)$ is to pick a sufficiently small ample ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $A$ on $X$, because then one knows that $\mathbf{B}_+(D) = \mathbf{B}(D-A)$ by \cite[Prop. 1.5]{elmnp1}.
In the case when $D$ is also nef, for every subvariety $V \subseteqset X$ of dimension $d \geq 1$ such that $D^d \cdot V = 0$, we have that $D_{|V}$ is not big, whence $s(D-A)_{|V}$ cannot be effective for any $s ^{-1}n {\mathbb N}$ and therefore $V \subseteqseteq \mathbf{B}(D-A) = \mathbf{B}_+(D)$. Now define
\[ {\rm Null}(D) = \bigcup_{V \subseteqset X: D^d \cdot V = 0} V \]
so that, by what we just said,
\begin{equation}
\label{nak}
{\rm Null}(D) \subseteqseteq \mathbf{B}_+(D).
\end{equation}
A somewhat surprising result of Nakamaye \cite[Thm. 0.3]{n} (see also \cite[\S 10.3]{laz}) asserts that, if $X$ is {^{-1}t smooth} and $D$ is big and nef, then in fact equality holds in \eqref{nak}.
As is well-known, in birational geometry, one must work with normal varieties with some kind of (controlled) singularities. In the light of this, it becomes apparent that it would be nice to have a generalization of Nakamaye's Theorem to normal varieties. While in positive characteristic the latter has been recently proved to hold, on any projective scheme, by Cascini, McKernan and Musta{\c{t}}{\u{a}} \cite[Thm. 1.1]{cmm}, we will show in this article a generalization to normal complex varieties with log canonical singularities. This partially answers a question in \cite{cmm}.
More precisely let us define
\begin{defn}
{\rm Let $X$ be a normal projective variety. The {\bf non-lc locus} of $X$ is
\[ X_{\mathrm{nlc}} =\bigcap_{{\Delta}} \mathrm{Nlc}(X,{\Delta}) \]
where ${\Delta}$ runs among all effective Weil ${\mathbb Q}$-divisors such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})$ is the locus of points $x ^{-1}n X$ such that $(X, {\Delta})$ is not log canonical at $x$.}
\end{defn}
Using Ambro's and Fujino's theory of non-lc ideal sheaves \cite{a}, \cite{f} and a modification of some results of de Fernex and Hacon \cite{dh}, we prove
^{\vee}skip .3cm
\begin{bthm}
\label{main}
Let $X$ be a normal projective variety such that $\dim X_{\mathrm{nlc}} \leq 1$.
Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bthm}
^{\vee}skip .2cm
This easily gives the following
\begin{bcor}
\label{dim}
Let $X$ be a normal projective variety such that $\dim \operatorname{Sing}(X) \leq 1$ or $\dim X \leq 3$ or there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ such that $(X, {\Delta})$ is log canonical.
Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bcor}
^{\vee}skip .2cm
Moreover, using a striking result of Gibney, Keel and Morrison \cite[Thm. 0.9]{gkm}, we can give a very quick application to the moduli space of stable pointed curves.
\begin{bcor}
\label{mod}
Let $g \geq 1$ and let $D$ be a big and nef ${\mathbb Q}$-divisor on $\overline{M}_{g,n}$. Then
\[ \mathbf{B}_+(D) \subseteqseteq \partial \overline{M}_{g,n}. \]
\end{bcor}
Thus, for example, one gets new compactifications of $M_{g,n}$ by taking rational maps associated to such divisors.
The second base locus associated to any pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor $D$, measuring how far $D$ is from being nef, is the restricted base locus \cite[Def. 1.12]{elmnp1}.
\begin{defn}
{\rm Let $X$ be a normal projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. The {\bf restricted base locus} of $D$ is
\[ \mathbf{B}_-(D)= \bigcup_{A {\ \rm ample}} \mathbf{B}(D+A) \]
where $A$ runs among all ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors such that $D+A$ is a ${\mathbb Q}$-divisor.}
\end{defn}
Restricted base loci are countable unions of subvarieties by \cite[Prop. 1.19]{elmnp1}, but not always closed \cite[Thm. 1.1]{les}.
For a big ${\mathbb Q}$-divisor $D$ on a {^{-1}t smooth} variety $X$, the subvarieties of $\mathbf{B}_-(D)$ are precisely described in \cite[Prop. 2.8]{elmnp1} (also in positive characteristic in \cite[Thm. 6.2]{mus}) in terms of asymptotic valuations.
\begin{defn}
\label{as1}
{\rm (\cite[Def. III.2.1]{nak}, \cite[Lemma 3.3]{elmnp1}, \cite[\S 1.3]{bbp}, \cite[\S 2]{dh})
Let $X$ be a normal projective variety, let $D$ be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$, that is $v$ is a positive integer multiple of the valuation associated to a prime divisor $\Gamma$ lying on a birational model $f : Y \to X$. The center of $v$ on $X$ is $c_X(v)=f(\Gamma)$.
If $D$ is big, we set
\[ v(\|D\|) = ^{-1}nf \{v(E), E \ \mbox{effective} \ {\mathbb R}\mbox{-Cartier} \ {\mathbb R}\mbox{-divisor on X such that} \ E \equiv D\}; \]
if $D$ is pseudoeffective, we pick an ample divisor $A$ and set
\[ v(\|D\|) = \lim_{^{\vee}arepsilon \to 0^+} v(\|D + ^{\vee}arepsilon A\|). \]
If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and $b ^{-1}n {\mathbb N}$ is such that $bD$ is Cartier and $|bD| \neq \emptyset$, we set
(see \cite[Def. 2.14]{cd} or \cite[Def. 2.2]{elmnp1} for the case $D$ big)
\[ v(\langle D \rangle) = \lim_{m \to + ^{-1}nfty}\frac{v(|mbD|)}{mb} \]
where, if $g$ is an equation, at the generic point of $c_X(v)$, of a general element in $|mbD|$, then $v(|mbD|) = v(g)$.}
\end{defn}
Now the main content of \cite[Prop. 2.8]{elmnp1} is that, given a discrete valuation $v$ on a {^{-1}t smooth} $X$ with center $c_X(v)$ and a big divisor $D$, then $c_X(v) \subseteqseteq \mathbf{B}_-(D)$ if and only if $v(\|D\|) > 0$. Using the main result of \cite{cd} we give a generalization to normal pairs with klt singularities.
\begin{bthm}
\label{main2}
Let $X$ be a normal projective variety such that there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ with $(X,{\Delta})$ a klt pair. Let $v$ be a divisorial valuation on $X$.
Then
\begin{itemize}
^{-1}tem [(i)] If $D$ is a big Cartier divisor on $X$ we have
\[ v(\langle D \rangle) > 0 \ \mbox{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D) \ \mbox{if and only if} \ \limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty. \]
^{-1}tem [(ii)] If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
\[ v(\|D\|) > 0 \ \textrm{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D). \]
\end{itemize}
\end{bthm}
^{\vee}skip .2cm
\noindent {^{-1}t Acknowledgments}. We wish to thank Lorenzo Di Biagio for some helpful discussions. | 3,192 | 15,835 | en |
train | 0.30.1 | \section{Non-lc ideal sheaves}
\label{basic}
\begin{notn}
{\rm Throughout the article we work over the complex numbers. Given a variety $X$ and a coherent sheaf of ideals $\mathcal{J} \subseteqset \mathcal{O}_X$, we denote by $\mathcal{Z}(\mathcal{J})$ the closed subscheme of $X$ defined by $\mathcal{J}$. If $X$ is a normal projective variety and ${\Delta}$ is a Weil ${\mathbb Q}$-divisor on $X$, we call $(X, {\Delta})$ a {\bf pair} if $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier. We refer to \cite[Def. 2.34]{km} for the various notions of singularities of pairs.}
\end{notn}
\begin{defn}
{\rm Let $X$ be a normal projective variety and let ${\Delta} = \sum\limits_{i=1}^s d_i D_i$ be a Weil ${\mathbb Q}$-divisor on $X$, where the ${D_i}'s$ are distinct prime divisors.
Given $a ^{-1}n {\mathbb R}$ we set ${\Delta}^{>a}=\sum\limits_{1 \leq i \leq s : d_i>a} d_i D_i$, ${\Delta}^+={\Delta}^{>0}$, ${\Delta}^- =(-{\Delta})^+$ and ${\Delta}^{<a}=-((-{\Delta})^{>-a})$. The {\bf round up} of ${\Delta}$ is $\lceil {\Delta} \rceil = \sum\limits_{i=1}^s \lceil d_i \rceil D_i$ and the {\bf round down} is $\lfloor {\Delta} \rfloor = \sum\limits_{i=1}^s \lfloor d_i \rfloor D_i$. We also set ${\Delta}^{\#} = {\Delta}^{<-1} + {\Delta}^{>-1}$.}
\end{defn}
The following is easily proved.
\begin{rem}
\label{sha}
{\rm Let $X$ be a normal projective variety and let ${\Delta}, {\Delta}'$ be Weil ${\mathbb Q}$-divisors on $X$. Then
\begin{itemize}
^{-1}tem[(i)] $\lceil (-{\Delta})^{\#} \rceil = \lceil -({\Delta}^{<1}) \rceil - \lfloor {\Delta}^{>1} \rfloor$;
^{-1}tem[(ii)] If ${\Delta} \leq {\Delta}'$, then $\lceil {\Delta}^{\#} \rceil \leq \lceil ({\Delta}')^{\#} \rceil$.
\end{itemize}}
\end{rem}
We recall the definition of non-lc ideal sheaves \cite[Def. 4.1]{a}, \cite[Def. 2.1]{f}.
\begin{defn}
\label{inlc}
{\rm Let $(X, {\Delta})$ be a pair and let $f: Y \to X$ be a resolution of $X$ such that ${\Delta}_Y :=f^{\ast}(K_X+{\Delta})-K_Y$ has simple normal crossing support.
The {\bf non-lc ideal sheaf associated to} $(X,{\Delta})$ is
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor). \]}
\end{defn}
\begin{rem}
\label{nlc}
{\rm Non-lc ideal sheaves are well-defined by \cite[Prop. 2.6]{f}, \cite[Rmk. 4.2(iv)]{a}. Moreover, when ${\Delta}$ is effective and $f : Y \to X$ is a log-resolution of $(X, {\Delta})$, we have that the non-lc locus of $(X, {\Delta})$ is, set-theoretically, $\operatorname{Nlc}(X,{\Delta})= f(\operatorname{Supp}({\Delta}_Y^{>1})) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))$ \cite[Lemma 2.2]{f}.}
\end{rem}
\begin{rem}
\label{ic}
{\rm The non-lc ideal sheaf of a pair $(X,{\Delta})$ with ${\Delta}$ effective is an integrally closed ideal.}
\begin{proof}
With notation as in Definition \ref{inlc}, set $G = \lceil - ({\Delta}_Y^{<1}) \rceil$ and $N= \lfloor {\Delta}_Y^{>1} \rfloor$, so that $G$ and $N$ are effective divisors without common components, $G$ is $f$-exceptional and $\mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(G-N) = f_{\ast}\mathcal{O}_Y(-N)$ by Fujita's lemma \cite[Lemma 2.2]{fta}, \cite[Lemma 1-3-2]{kmm}, \cite[Lemma 4.5]{dh}. Therefore $\mathcal{J}_{NLC}(X,{\Delta})$ is an ideal sheaf and it is integrally closed by \cite[Prop. 9.6.11]{laz}.
\end{proof} | 1,283 | 15,835 | en |
train | 0.30.2 | \end{rem}
Our next goal is to prove, using techniques and results in de Fernex-Hacon \cite{dh}, that non-lc ideal sheaves have a unique maximal element. To this end we will use some results of Fujino \cite{f} and de Fernex-Hacon \cite{dh} that we wish to recall for the reader's convenience.
\begin{lemma} \cite[Lemma 2.7]{f}
\label{fuj}
Let $g : Y' \to Y$ be a proper birational morphism between smooth varieties and let $B_Y$ be an ${\mathbb R}$-divisor on $Y$ having simple normal crossing support. Assume that $B_{Y'} := g^{\ast}(K_Y+B_Y)-K_{Y'}$ also has simple normal crossing support. Then
\[ g_{\ast} {\mathcal O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) \cong {\mathcal O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ). \]
\end{lemma}
\begin{defn}
\label{dfh1}
{\rm (\cite[Def. 3.1]{dh})
Let $f: Y \to X$ be a proper birational morphism between normal varieties and let $K_Y$ be a canonical divisor on $Y$ and $K_X = f_{\ast} K_Y$. For every $m \geq 1$ define $K_{m, Y/X} = K_Y - \frac{1}{m}f^{\natural}(mK_X)$, where $f^{\natural}(mK_X)$ is the divisor on $Y$ such that ${\mathcal O}_Y(-f^{\natural}(mK_X)) = ({\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y)^{^{\vee}ee ^{\vee}ee}$.}
\end{defn}
\begin{lemma}
\label{dh2}
Let $m \geq 1$. In (i)-(iv) below let $f: Y \to X$ be a proper birational morphism between normal varieties. Then
\begin{itemize}
^{-1}tem [(i)] If $X$ is Gorenstein then $K_{m, Y/X} = K_{Y/X} := K_Y + f^{\ast}(-K_X)$;
^{-1}tem [(ii)] \cite[Rmk 3.3]{dh} For all $q \geq 1$ we have $K_{m, Y/X} \leq K_{mq, Y/X}$;
^{-1}tem [(iii)] \cite[Lemma 3.5]{dh} Assume that $mK_Y$ is Cartier and ${\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y$ is invertible. Let $Y'$ be a normal variety and let $g : Y' \to Y$ be a proper birational morphism. Then $K_{m, Y'/X} = K_{m, Y'/Y} + g^{\ast} K_{m, Y/X}$;
^{-1}tem [(iv)] \cite[Rmk 3.9]{dh} Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and assume that $m(K_X + {\Delta}elta)$ is Cartier. Then $K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{m, Y/X}$;
^{-1}tem [(v)] \cite[Thm. 5.4 and its proof]{dh} For every $m \geq 2$ there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ such that $m{\Delta}_m$ is integral, $\lfloor {\Delta}_m \rfloor = 0$, $({\Delta}_m)_Y$ has simple normal crossing support, $f$ is a log-resolution for the log-pair $((X, {\Delta}_m), \mathcal{O}_X(-mK_X))$, $K_X + {\Delta}_m$ is ${\mathbb Q}$-Cartier and $K_{m, Y/X} = K_Y + f_{\ast}^{-1}({\Delta}_m) - f^{\ast}(K_X+{\Delta}_m)$.
\end{itemize}
\end{lemma}
In (iv) and (v) above $f_{\ast}^{-1}({\Delta})$ is the proper transform of ${\Delta}$. Note that our ${\Delta}_Y$ (Definition \ref{inlc}) is different from the one in \cite[Def. 3.8]{dh}.
Now we have
\begin{prop}
\label{max}
Let $X$ be a normal projective variety. Then there exists a Weil ${\mathbb Q}$-divisor ${\Delta}_0$ on $X$ such that $\lfloor {\Delta}_0 \rfloor = 0$,
$K_X + {\Delta}_0$ is ${\mathbb Q}$-Cartier and
\[ \mathcal{J}_{NLC}(X,{\Delta}) \subseteqseteq \mathcal{J}_{NLC}(X,{\Delta}_0) \]
for every pair $(X,{\Delta})$ with ${\Delta}$ effective.
\end{prop}
\begin{proof}
Fix a canonical divisor $K_X$ on $X$ and an integer $m \geq 2$. By Lemma \ref{dh2}(v) there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ with the properties in (v). In particular $K_{m, Y/X}$ is $f$-exceptional. Now set
\[ \mathfrak{a}_m(X) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil).\]
As in the proof of Remark \ref{ic} we get that $\mathfrak{a}_m(X)$ is a coherent ideal sheaf. Let us check that its definition is independent of the choice of $f$. Let $f': Y' \to X$ be another log-resolution of $(X, \mathcal{O}_X(-mK_X))$ and assume, as we may, that $f'$ factors through $f$ and a morphism $g : Y' \to Y$. By Lemma \ref{dh2}(iii) and (i) we have
$K_{m, Y'/X} = K_{Y'/Y} + g^{\ast} K_{m, Y/X} = K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X})$, whence
\begin{equation}
\label{ben}
(fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast}(g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil)).
\end{equation}
Now set $B_Y = - K_{m, Y/X}$ and $B_{Y'} = g^{\ast}(K_Y+B_Y) - K_{Y'}$ so that, using Remark \ref{sha}(i) and Lemma \ref{fuj}, we have
\[ g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil (-B_{Y'})^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) = \]
\[ = \mathcal{O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ) = \mathcal{O}_Y (\lceil (-B_Y)^{\#} \rceil) = \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]
and by \eqref{ben} we get
\[ (fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast} \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]
that is $\mathfrak{a}_m(X)$ is well defined.
We now claim that the set $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element. In fact, given $m, q \geq 2$, let $f: Y \to X$ be a log-resolution of $(X, \mathcal{O}_X(-mK_X)) + \mathcal{O}_X(-mqK_X))$. By Lemma \ref{dh2}(ii) and Remark \ref{sha}(ii) we have $\lceil (K_{m, Y/X})^{\#} \rceil \leq \lceil (K_{mq, Y/X})^{\#} \rceil$ and therefore $\mathfrak{a}_m(X) \subseteqseteq \mathfrak{a}_{mq}(X)$. Using the ascending chain condition on ideals we conclude that $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element, which we will denote by $\mathfrak{a}_{\rm max}(X)$.
Next let us show that all the ideal sheaves $\mathfrak{a}_m(X)$, for $m \geq 2$ (whence in particular also $\mathfrak{a}_{\rm max}(X)$), are in fact non-lc ideal sheaves of a suitable pair.
Let ${\Delta}_m$ be as above, so that, by Remark \ref{sha}(i) and using $\lceil (-f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = 0$, we have
\[ \lceil - (({\Delta}_m)_Y^{<1}) \rceil - \lfloor ({\Delta}_m)_Y^{>1} \rfloor = \lceil (-({\Delta}_m)_Y)^{\#} \rceil = \lceil (K_{m, Y/X} - f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \]
\[ = \lceil (K_{m, Y/X})^{\#} \rceil + \lceil (- f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \lceil (K_{m, Y/X})^{\#} \rceil \]
whence
\[ \mathcal{J}_{NLC}(X,{\Delta}_m)=f_{\ast}\mathcal{O}_Y(\lceil - (({\Delta}_m)_Y^{<1}) \rceil - \lfloor ({\Delta}_m)_Y^{>1} \rfloor) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) = \mathfrak{a}_m(X). \]
To finish the proof, let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $q ^{-1}n {\mathbb N}$ be such that $q(K_X + {\Delta})$ is Cartier. Let $m_0 \geq 2$ be such that $\mathfrak{a}_{\rm max}(X) = \mathfrak{a}_{m_0}(X) = \mathfrak{a}_{qm_0}(X)$. By what we proved above, there exists ${\Delta}_0 := {\Delta}_{qm_0}$ such that $\mathcal{J}_{NLC}(X,{\Delta}_0) = \mathfrak{a}_{\rm max}(X)$. By Lemma \ref{dh2}(iv) we have that $- {\Delta}_Y \leq K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{qm_0, Y/X}$, whence also, by Remark \ref{sha} (i) and (ii),
\[ \lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor = \lceil (- {\Delta}_Y)^{\#} \rceil \leq \lceil (K_{qm_0, Y/X})^{\#} \rceil \]
and therefore
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor) \subseteqseteq f_{\ast}\mathcal{O}_Y(\lceil (K_{qm_0, Y/X})^{\#} \rceil) = \mathfrak{a}_{\rm max}(X) = \mathcal{J}_{NLC}(X,{\Delta}_0). \]
\end{proof} | 3,131 | 15,835 | en |
train | 0.30.3 | \section{Proof of Theorem \ref{main}}
\label{b+}
We record the following lemma, which is also of independent interest.
\begin{lemma}
\label{bs}
Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $D$ be an effective Cartier divisor on $X$. Then there exists $c = c(X,{\Delta}, D) ^{-1}n {\mathbb N}$ such that the set-theoretic equality
\[ \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E_1+\dots+E_c)) \]
holds for some $E_1,\dots, E_c ^{-1}n |D|$.
\end{lemma}
\begin{proof}
Let $f:Y \to X$ be a log-resolution of $(X,{\Delta})$ and of the linear series $|D|$ such that $f_{\ast}^{-1} {\Delta} + \mathbf{B}s|f^{\ast}D| + \operatorname{Exc}(f)$ has simple normal crossing support.
Write ${\Delta}_Y = {\Delta}_Y^+ - {\Delta}_Y^-$, where ${\Delta}_Y^+$ and ${\Delta}_Y^-$ are effective simple normal crossing support ${\mathbb Q}$-divisors without common components. Then ${\Delta}_Y^-= \sum_{i=1}^s \delta_i D_i$, for some non-negative $\delta_i ^{-1}n {\mathbb Q}$ and distinct prime divisors $D_i$'s and define
\[ c = \lceil \operatorname{max}\{\delta_i, 1 \leq i \leq s \} \rceil +2. \]
Moreover we have that $|f^{\ast}D|=|M|+F$, where $|M|$ is base-point free and $\operatorname{Supp}(F)= \mathbf{B}s|f^{\ast}D|$. By Bertini's Theorem and \cite[Lemma 9.1.9]{laz}, we can choose $M_1,\dots, M_c ^{-1}n |M|$ general divisors such that, for all $j=1,\dots,c$, $M_j$ is smooth, every component of $M_j$ is not a component of ${\Delta}_Y, M_1,\dots, M_{j-1}$ and ${\Delta}_Y+M_1+\dots+M_c+F$ has simple normal crossing support. Now, for all $j = 1, \ldots, c$, $M_j+F ^{-1}n |f^{\ast}D|$, so that there exists $E_j ^{-1}n |D|$ such that $M_j+F =f^{\ast}E_j$. Set $E=E_1+\dots +E_c$ and notice that $f$ is also a log-resolution of $(X,{\Delta}+E)$.
By Remark \ref{nlc} we have $\operatorname{Nlc}(X,{\Delta})=\mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))\subseteqseteq \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$, the latter inclusion following by Remark \ref{sha}(i) and (ii), because $E$ is effective. Also, for every prime divisor $\Gamma$ in the support of $F$ we get for the discrepancies
\[ a(\Gamma,X,{\Delta}+E)=a(\Gamma,X,{\Delta})-\operatorname{ord}_{\Gamma}(f^{\ast} E) =-\operatorname{ord}_{\Gamma} ({\Delta}_Y) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq \]
\[ \leq \operatorname{ord}_{\Gamma} ({\Delta}_Y^-) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq \operatorname{max}\{\delta_i, 1 \leq i \leq s \} -\operatorname{ord}_{\Gamma}(M_1+\dots+M_c+cF)\leq -2 \]
whence $f(\Gamma) \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E)$. As $\mathbf{B}s|D|$ is the union of such $f(\Gamma)$'s, using Remark \ref{nlc}, we get the inclusion $\mathbf{B}s |D| \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$.
On the other hand notice that $({\Delta}+E)_Y =f^{\ast}(K_X+{\Delta}+E)-K_Y= {\Delta}_Y +f^{\ast}E$. Also ${\Delta}_Y + f^{\ast} E = {\Delta}_Y + M_1+\dots+M_c+cF$, so that
\[ \operatorname{Supp}(({\Delta}+E)_Y^{>1}) = \operatorname{Supp}(({\Delta}_Y + f^{\ast} E)^{>1}) \subseteqseteq \operatorname{Supp}(F) \cup \operatorname{Supp} ({\Delta}_Y^{>1}) \]
whence
\[ f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq f(\operatorname{Supp}(F)) \cup f(\operatorname{Supp} ({\Delta}_Y^{>1})) = \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
Therefore, by Remark \ref{nlc},
\[ \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) = \operatorname{Nlc}(X, {\Delta}+E) = f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
\end{proof}
Now we essentially follow the proof of Nakamaye's Theorem as in \cite[\S 10.3]{laz} and \cite[Thm. 0.3]{n}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main}}}
\begin{proof}
We can assume that $D$ is a Cartier divisor. The issue is of course to prove that $\mathbf{B}_+(D) \subseteqseteq \operatorname{Null}(D)$, since the opposite inclusion holds on any normal projective variety, as explained in the introduction.
By Proposition \ref{max} and Remark \ref{nlc} there is an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ on $X$ such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})= X_{\mathrm{nlc}}$, so that $\dim \mathrm{Nlc}(X,{\Delta}) \leq 1$.
Let $A$ be an ample Cartier divisor such that $A-(K_X+{\Delta})$ is ample. As in \cite[Proof of Thm. 10.3.5]{laz}) we can choose $a, p ^{-1}n {\mathbb N}$ sufficiently large such that
\[ \mathbf{B}_+(D)= \mathbf{B}(aD-2A)= \mathbf{B}s|paD-2pA|. \]
By Lemma \ref{bs} there exist $c ^{-1}n {\mathbb N}$ and a Cartier divisor $E$ on $X$ such that
\[ \mathbf{B}_+(D)\cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) \]
and $E \equiv c(paD-2pA)=qaD-2qA$, where $q:= cp ^{-1}n {\mathbb N}$.
Set $Z = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$. For $m \geq qa$, we get that
\[ mD-qA-(K_X+{\Delta}+E)\equiv (m-qa)D+qA-(K_X+{\Delta}) \]
is ample, whence $H^1(X, \mathcal{J}_{NLC}(X,{\Delta}+E) \otimes \mathcal{O}_X(mD-qA))=0$, for $m\geq qa$
by \cite[Thm. 3.2]{f}, \cite[Thm. 4.4]{a}, so that the restriction map
\begin{equation}
\label{uno}
H^0(X,\mathcal{O}_X(mD-qA))\to H^0(Z, \mathcal{O}_Z(mD-qA)) \mbox{\ is surjective for} \ m \geq qa.
\end{equation}
By contradiction let us assume that there exists an irreducible component $V$ of $\mathbf{B}_+(D)$, such that $V \not \subseteqseteq \operatorname{Null}(D)$.
Now $V \subseteqseteq \mathbf{B}_+(D) \subseteqseteq \mathbf{B}(D-\frac{q}{m}A)\subseteqseteq \mathbf{B}s |mD-qA|$ for $m ^{-1}n {\mathbb N}$, whence the restriction map
\[ H^0(X,\mathcal{O}_X(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m ^{-1}n {\mathbb N} \]
and therefore, by \eqref{uno}, also
\begin{equation}
\label{due}
H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m \geq qa.
\end{equation}
On the other hand $\dim V\geq 1$, as $\mathbf{B}_+(D)$ does not contain isolated points by \cite[Proposition 1.1]{elmnp2}(which holds on $X$ normal).
As $\dim \operatorname{Nlc}(X,{\Delta}) \leq 1$, this implies that $V$ is an irreducible component of $Z$. Moreover, as $V\not \subseteqseteq \operatorname{Null}(D)$, we have that $D_{|_V}$ is big.
Now, by Remark \ref{ic}, $\mathcal{J}_{NLC}(X,{\Delta}+E)$ is integrally closed, and exactly as in \cite[Proof of Thm. 10.3.5]{laz} (the proof of this part holds on any normal projective variety) it follows that, for $m \gg 0$, $H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA))$ is not zero, thus contradicting \eqref{due}.
This concludes the proof.
\end{proof}
\renewcommand{Proof}{Proof}
\renewcommand{Proof}{Proof of Corollary {\rm \ref{dim}}}
\begin{proof}
Note that, on any normal projective variety $X$, we have $X_{\mathrm{nlc}} \subseteqseteq \operatorname{Sing}(X)$ (see for example \cite[Rmk 4.8]{cd}) and if $\dim X\leq 3$, then $\dim \operatorname{Sing}(X) \leq 1$. Then just apply Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof}
\renewcommand{Proof}{Proof of Corollary {\rm \ref{mod}}}
\begin{proof}
By \cite[Thm. 0.9]{gkm} we know that $\operatorname{Null}(D) \subseteqseteq \partial \overline{M}_{g,n}$. On the other hand it is well-known
(see for example \cite[Lemma 10.1]{bchm}) that $(\overline{M}_{g,n}, 0)$ is klt, whence the conclusion follows by Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof} | 2,847 | 15,835 | en |
train | 0.30.4 | \section{Restricted base loci on klt pairs}
\label{rests}
We first recall that, associated to a pseudoeffective divisor $D$, there are two more loci, one that also measures how far $D$ is from being nef and another one that measures how far $D$ is from being nef and abundant.
\begin{defn}
\label{as2}
{\rm Let $X$ be a normal projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. As in \cite[Def. 1.7]{bbp}, we define the {\bf non-nef locus}
\[ \operatorname{Nnef}(D) = \bigcup_{v : v(\|D\|) > 0} c_X(v) \]
where $v$ runs among all divisorial valuations on $X$, $c_X(v)$ is its center and $v(\|D\|)$ is as in Definition \ref{as1}.
Let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$. As in \cite[Def. 2.18]{cd}, we define the {\bf non nef-abundant locus}
\[ \operatorname{Nna}(D) = \bigcup_{v : v(\langle D \rangle) > 0} c_X(v) \]
where again $v$ runs among all divisorial valuations on $X$ and $v(\langle D \rangle)$ is as in Definition \ref{as1}.}
\end{defn}
In the sequel we will use the fact that, for $D$ big (\cite[Lemma 3.3]{elmnp1}) or even abundant (\cite[Prop. 6.4]{leh}), we have $v(\|D\|) = v(\langle D \rangle)$, while in general they are different when $D$ is only pseudoeffective (\cite[Rmk 2.16]{cd}).
We will also use (see \cite[page 2]{bfj} and references therein)
\noindent {\bf Izumi's Theorem}
\label{iz}
{^{-1}t Let $X$ be a normal variety over an algebraically closed field $k$ and let $0 ^{-1}n X$ be a closed point. Let $m_0$ be the maximal ideal of the local ring ${\mathcal O}_{X,0}$ and set, for any $f ^{-1}n {\mathcal O}_{X,0}$, $\operatorname{ord}_0 (f) = \operatorname{max}\{j \geq 0 : f ^{-1}n m_0^j \}$.
For any divisorial valuation $v$ of $k(X)$ centered at $0$, there exists a constant $C = C(v) > 0$ such that
\[ C^{-1} \operatorname{ord}_0(f) \leq v(f) \leq C \operatorname{ord}_0(f).\]}
We start by proving an analogue of \cite[Prop. 2.8]{elmnp1} for $\operatorname{Nna}(D)$.
\begin{thm}
\label{nna}
Let $X$ be a normal projective variety, let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and let $v$ be a divisorial valuation on $X$.
Then
\[ c_X(v)\subseteqseteq \operatorname{Nna}(D) \ \textrm{if and only if} \ v(\langle D \rangle)>0. \]
\end{thm}
\begin{proof}
We can assume that $D$ is Cartier and effective. By definition of $\operatorname{Nna}(D)$, we just need to prove that if $c_X(v) \subseteqseteq \operatorname{Nna}(D)$, then $v(\langle D \rangle)>0$.
We first prove the theorem when $X$ is smooth. For any $p ^{-1}n {\mathbb N}$ let $b(|pD|)$ be the base ideal of $|pD|$, $\mathcal{J}(X,\|pD\|))$ the asymptotic multiplier ideal and denote by $b_p$ and $j_p$ the corresponding images in $R_v$, the DVR associated to $v$. As in \cite[\S 2]{elmnp1}, we get
\begin{equation}
\label{j}
v(\langle D \rangle) = \lim_{p \to +^{-1}nfty} \frac{v(b_p)}{p} \geq \lim_{p \to +^{-1}nfty} \frac{v(j_p)}{p} = \sup_{p ^{-1}n {\mathbb N}} \{\frac{v(j_p)}{p}\}.
\end{equation}
By \cite[Cor. 5.2]{cd} we have the set-theoretic equality
\[ \operatorname{Nna}(D)=\bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}(X,\|pD\|)) \]
whence there exists $p_0 ^{-1}n {\mathbb N}$ such that $c_X(v) \subseteqseteq \mathcal{Z}(\mathcal{J}(X,\|p_0D\|))$, so that $v(j_{p_0})>0$ and \eqref{j} gives that $v(\langle D \rangle)>0$.
We now prove the theorem for a divisorial valuation $\nu$ on $X$ such that $c_X(\nu) = \{x\}$ is a closed point.
As $c_X(\nu) \subseteqseteq \operatorname{Nna}(D)$, there exists a divisorial valuation $v_0$ on $X$ such that $v_0(\langle D \rangle)>0$ and $x ^{-1}n c_X(v_0)$.
Let $E_0$ be a prime divisor over $X$ such that $v_0 = k \operatorname{ord}_{E_0}$ for some $k ^{-1}n {\mathbb N}$. We can assume that there is a birational morphism $\mu:Y\to X$ from a smooth
variety $Y$ such that $E_0 \subseteqset Y$. As $\mu(E_0)=c_X(\operatorname{ord}_{E_0}) = c_X(v_0 )$, there is a point $y ^{-1}n E_0$ such that $\mu(y) = x$. Let $\pi : Y' \to Y$ be the blow-up of $Y$ on $y$ with exceptional divisor $E_y$. For any $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$$\operatorname{ord}_{E_y}(G) =\operatorname{ord}_{E_y}(\pi^*(\mu^*G)) = \operatorname{ord}_{y}(\mu^*G) \geq \operatorname{ord}_{E_0}(\mu^*G) = \operatorname{ord}_{E_0}(G)$$
therefore $\operatorname{ord}_{E_y}(\langle D \rangle) \geq \operatorname{ord}_{E_0}(\langle D \rangle) = \frac{1}{k} v_0(\langle D \rangle)>0$.
Since $c_X(\operatorname{ord}_{E_y})=\{x\}$, by Izumi's Theorem applied twice, there exist $C > 0, C' > 0$ such that for all $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$\operatorname{ord}_{E_y}(G) \leq C' \operatorname{ord}_{x}(G) \leq C \nu(G)$. Hence $\nu(\langle D \rangle) \geq \frac{1}{C} \operatorname{ord}_{E_y}(\langle D \rangle)>0$.
Finally let $v$ be any divisorial valuation on $X$ with $c_X(v)\subseteqseteq \operatorname{Nna}(D)$. As above there is a birational morphism $f : Z \to X$ from a smooth
variety $Z$ and a prime divisor $E \subseteqset Z$ such that $v = h \operatorname{ord}_E$ for some $h ^{-1}n {\mathbb N}$. For every closed point $z ^{-1}n E$ we have that $\nu:= \operatorname{ord}_z$ is a divisorial valuation with $c_X(\nu) \subseteqseteq c_X(\operatorname{ord}_E) \subseteqseteq \operatorname{Nna}(D)$ and $c_X(\nu)$ is a closed point. Thus, by what we proved above, we have that $\operatorname{ord}_z(\langle f^*(D) \rangle) = \operatorname{ord}_z(\langle D \rangle)>0$ for all $z ^{-1}n E$, so that $E \subseteqseteq \operatorname{Nna}(f^*(D))$. As $Z$ is smooth, we get $v(\langle D \rangle) = h\operatorname{ord}_E(\langle D \rangle) = h \operatorname{ord}_E(\langle f^*(D) \rangle)>0$.
\end{proof}
We next prove an analogous result for $\operatorname{Nnef}(D)$.
\begin{thm}
\label{nnef}
Let $X$ be a normal projective variety, let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$.
Then
$$c_X(v) \subseteqseteq \operatorname{Nnef}(D) \ \textrm{if and only if} \ v(\|D\|)>0.$$
\end{thm}
\begin{proof} Again we need to prove that $v(\|D\|) > 0$ if $c_X(v) \subseteqseteq \operatorname{Nnef}(D)$.
By \cite[Lemmas 2.13 and 2.12]{cd}, there exists a sequence of ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors $\{A_m\}_{m ^{-1}n {\mathbb N}}$
such that $\|A_m\| \to 0$, $D+A_m$ is a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor for all $m ^{-1}n {\mathbb N}$ and
$$\operatorname{Nnef}(D)=\bigcup_{m^{-1}n {\mathbb N}} \operatorname{Nnef}(D+A_m).$$
Then there is $m_0 ^{-1}n {\mathbb N}$ such that $c_X(v)\subseteqseteq \operatorname{Nnef}(D+A_{m_0})$. As $D+A_{m_0}$ is big, we have $\operatorname{Nnef}(D+A_{m_0}) = \operatorname{Nna}(D+A_{m_0})$, whence $v(\|D+A_{m_0}\|) = v(\langle D+A_{m_0}\rangle) >0$ by Theorem \ref{nna}.
Therefore $0 < v(\|D+A_{m_0}\|) \leq v(\|D\|) + v(\|A_{m_0}\|) = v(\|D\|)$.
\end{proof}
\begin{rem}
{\rm Note that, given a normal projective variety $X$, Theorems {\rm \ref{nna}} and {\rm \ref{nnef}} can be rewritten as follows (where $x$ is a closed point).
If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$ such that $\kappa(D)\geq 0$, then
$$\operatorname{Nna}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\langle D\rangle)>0\}.$$
If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, then
$$\operatorname{Nnef}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\|D\|)>0\}.$$}
\end{rem}
Next we will prove Theorem {\rm \ref{main2}. We will use a singular version (see for example \cite[Def. 2.2]{cd}) of standard asymptotic multiplier ideal sheaves \cite[Ch. 11]{laz}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main2}}}
\begin{proof}
In both cases we have that $\operatorname{Nnef}(D) = \mathbf{B}_-(D)$ by \cite[Thm. 1.2]{cd}, whence also $\operatorname{Nna}(D) = \mathbf{B}_-(D)$ in case (i). Then (ii) follows by Theorem \ref{nnef} and the first equivalence in (i) by Theorem \ref{nna}. To complete the proof of (i) we need to show that if $\limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty$ then $v(\langle D\rangle) > 0$, the reverse implication being obvious. We will proceed similarly to \cite[Proof of Prop. 2.8]{elmnp1} and \cite[Proof of Lemma 4.1]{cd}. If $v(\langle D\rangle) = 0$, by what we just proved, we have that $c_X(v) \not\subseteqseteq \mathbf{B}_-(D)$ and, by \cite[Cor. 5.2]{cd}, we have the set-theoretic equality
\[ \mathbf{B}_-(D) = \bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}((X,{\Delta});\|pD\|)) \]
where $\mathcal{J}((X,{\Delta});\|pD\|)$ is as in \cite[Def. 2.2]{cd}. Therefore $c_X(v) \not\subseteqseteq \mathcal{Z}(\mathcal{J}((X,{\Delta}); \|pD\|))$ for any $p ^{-1}n {\mathbb N}$. Let $H$ be a very ample Cartier divisor such that $H - (K_X + {\Delta})$ is ample and let $n = \dim X$. By Nadel's vanishing theorem \cite[Thm. 9.4.17]{laz}, we deduce that $\mathcal{J}((X,{\Delta});\|pD\|) \otimes \mathcal{O}_X((n + 1)H + pD)$ is $0$-regular in the sense of Castelnuovo-Mumford, whence globally generated, for every $p ^{-1}n {\mathbb N}$, and therefore $c_X(v) \not\subseteqseteq \mathbf{B}s|(n+1)H + pD|$. On the other hand, as $D$ is big, there is $m_0 ^{-1}n {\mathbb N}$ such that $m_0 D \sim (n + 1)H + E$ for some effective Cartier divisor $E$. Hence, for any $m \geq m_0$, we get $v(|mD|) = v(|(m-m_0)D + (n + 1)H + E|) \leq v(|(m-m_0)D + (n + 1)H|) + v(|E|) = v(|E|)$ and the theorem follows.
\end{proof} | 3,744 | 15,835 | en |
train | 0.30.5 | \begin{rem}
{\rm Note that, given a normal projective variety $X$, Theorems {\rm \ref{nna}} and {\rm \ref{nnef}} can be rewritten as follows (where $x$ is a closed point).
If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$ such that $\kappa(D)\geq 0$, then
$$\operatorname{Nna}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\langle D\rangle)>0\}.$$
If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, then
$$\operatorname{Nnef}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\|D\|)>0\}.$$}
\end{rem}
Next we will prove Theorem {\rm \ref{main2}. We will use a singular version (see for example \cite[Def. 2.2]{cd}) of standard asymptotic multiplier ideal sheaves \cite[Ch. 11]{laz}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main2}}}
\begin{proof}
In both cases we have that $\operatorname{Nnef}(D) = \mathbf{B}_-(D)$ by \cite[Thm. 1.2]{cd}, whence also $\operatorname{Nna}(D) = \mathbf{B}_-(D)$ in case (i). Then (ii) follows by Theorem \ref{nnef} and the first equivalence in (i) by Theorem \ref{nna}. To complete the proof of (i) we need to show that if $\limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty$ then $v(\langle D\rangle) > 0$, the reverse implication being obvious. We will proceed similarly to \cite[Proof of Prop. 2.8]{elmnp1} and \cite[Proof of Lemma 4.1]{cd}. If $v(\langle D\rangle) = 0$, by what we just proved, we have that $c_X(v) \not\subseteqseteq \mathbf{B}_-(D)$ and, by \cite[Cor. 5.2]{cd}, we have the set-theoretic equality
\[ \mathbf{B}_-(D) = \bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}((X,{\Delta});\|pD\|)) \]
where $\mathcal{J}((X,{\Delta});\|pD\|)$ is as in \cite[Def. 2.2]{cd}. Therefore $c_X(v) \not\subseteqseteq \mathcal{Z}(\mathcal{J}((X,{\Delta}); \|pD\|))$ for any $p ^{-1}n {\mathbb N}$. Let $H$ be a very ample Cartier divisor such that $H - (K_X + {\Delta})$ is ample and let $n = \dim X$. By Nadel's vanishing theorem \cite[Thm. 9.4.17]{laz}, we deduce that $\mathcal{J}((X,{\Delta});\|pD\|) \otimes \mathcal{O}_X((n + 1)H + pD)$ is $0$-regular in the sense of Castelnuovo-Mumford, whence globally generated, for every $p ^{-1}n {\mathbb N}$, and therefore $c_X(v) \not\subseteqseteq \mathbf{B}s|(n+1)H + pD|$. On the other hand, as $D$ is big, there is $m_0 ^{-1}n {\mathbb N}$ such that $m_0 D \sim (n + 1)H + E$ for some effective Cartier divisor $E$. Hence, for any $m \geq m_0$, we get $v(|mD|) = v(|(m-m_0)D + (n + 1)H + E|) \leq v(|(m-m_0)D + (n + 1)H|) + v(|E|) = v(|E|)$ and the theorem follows.
\end{proof}
\renewcommand{Proof}{Proof}
We end the section with an observation on the behavior of these loci under birational maps.
\begin{cor}
Let $f:Y \to X$ be a projective birational morphism between normal projective varieties. Then:
\begin{itemize}
^{-1}tem [(i)] For every ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $D$ on $X$ such that $\kappa(D)\geq 0$, we have
$$\operatorname{Nna}(f^*(D))=f^{-1}(\operatorname{Nna}(D));$$
^{-1}tem [(ii)] For every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
$$\operatorname{Nnef}(f^*(D))=f^{-1}(\operatorname{Nnef}(D));$$
^{-1}tem [(iii)] If there exist effective Weil ${\mathbb Q}$-divisors ${\Delta}_X$ on $X$ and ${\Delta}_Y$ on $Y$ such that $(X,{\Delta}_X)$ and $(Y,{\Delta}_Y)$ are klt pairs,
then, for every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
$$\mathbf{B}_-(f^*(D))=f^{-1}(\mathbf{B}_-(D))$$
\end{itemize}
\end{cor}
\begin{proof}
To see (i), for every closed point $y ^{-1}n Y$, let $v_y$ be a divisorial valuation such that $c_Y(v_y)=\{y\}$.
Then, by Theorem \ref{nna}, we have,
\[ y^{-1}n f^{-1}(\operatorname{Nna}(D)){\mathcal L}eftrightarrow \{f(y)\}=c_X(v_y)\subseteqseteq \operatorname{Nna}(D){\mathcal L}eftrightarrow \]
\[ {\mathcal L}eftrightarrow v_y(\langle f^*(D) \rangle) = v_y(\langle D\rangle)>0{\mathcal L}eftrightarrow \{y\}=c_Y(v_y)\subseteqseteq \operatorname{Nna}(f^*(D)). \]
Now (ii) can be proved exactly in the same way by using Theorem \ref{nnef}, while (iii) follows from (ii) and \cite[Thm. 1.2]{cd}.
\end{proof}
\end{document} | 1,638 | 15,835 | en |
train | 0.31.0 | \begin{document}
\title{\Large \textbf{HALF-ISOMORPHISMS WHOSE INVERSES ARE ALSO HALF-ISOMORPHISMS}
\begin{abstract}
\noindent{}Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a \emph{half-isomorphism} if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$, for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism $f$ is called \emph{special} if $f^{-1}$ is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.
\end{abstract}
\noindent{}{\it Keywords}: half-isomorphism, half-automorphism, special half-isomorphism, groupoid, quasigroup, loop.
\section{Introduction}
A \emph{groupoid} consists of a nonempty set with a binary operation. A groupoid $( Q,*) $ is called a \emph{quasigroup} if for each $ a, b \in Q $ the equations $ a * x = b $ and $ y * a = b$ have unique solutions for $x,y \in Q$. A \textit{quasigroup} $(L,*)$ is a \emph{loop} if there exists an identity element $1\in L$ such that $ 1 * x = x = x * 1$, for any $ x \in L $. The fundamental definitions and facts from groupoids, quasigroups, and loops can be found in \cite{B71,P90}.
Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a \emph{half-isomorphism} if \mbox{$f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$,} for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a \emph{half-automorphism}. We say that a half-isomorphism is \emph{trivial} when it is either an isomorphism or an anti-isomorphism.
In 1957, Scott \cite{Sco57} showed that every half-isomorphism on groups is trivial. In the same paper, the author provided an example of a loop of order $8$ that has a nontrivial half-automorphism, then the result for groups can not be generalized to all loops. Recently, a similar version of Scott's result was proved for some subclasses of Moufang loops \cite{GG12,GGRS16,KSV16} and automorphic loops \cite{GA192}. A \emph{Moufang loop} is a loop that satisfies the identity $ x(y(xz))=((xy)x)z$, and an \emph{automorphic loop} is a loop in which every inner mapping is an automorphism \cite{BP56}. We note that there are Moufang loops and automorphic loops that have nontrivial half-automorphisms \cite{GG13,GA19,GPS17}.
In \cite{GA192}, the authors defined the concept of \emph{special half-isomorphism}. A half-isomorphism $f:G \rightarrow G'$ is called \emph{special} if the inverse mapping $f^{-1}:G' \rightarrow G$ is also a half-isomorphism. It is easy to construct an example of a
half-isomorphism that is not special, as we can see below.
\begin{exem}
\label{ex1} Let $G = \{1,2,...,6\}$ and consider the following Cayley tables of $(G,*)$ and $(G, \cdot)$:
\begin{center}
\begin{minipage}{.35\textwidth}
\centering
\begin{tabular}{c|cccccc}
$*$ & 1& 2& 3&4&5&6 \\
\hline
1 & 1&2 &3&4&5&6 \\
2 & 2&3 &4&5&6&1 \\
3 & 3&4 &5&6&1&2\\
4 & 4&5 &6&1&2&3 \\
5 & 5&6 &1&2&3&4 \\
6 & 6&1 &2&3&4&5 \\
\end{tabular}
\end{minipage}
\begin{minipage}{.35\textwidth}
\centering
\begin{tabular}{c|cccccc}
$\cdot $ & 1& 2& 3&4&5&6 \\
\hline
1 & 1&2 &3&4&5&6 \\
2 & 2&3 &4&5&6&1 \\
3 & 3&1 &5&6&4&2\\
4 & 4&5 &6&1&2&3 \\
5 & 5&6 &1&2&3&4 \\
6 & 6&4 &2&3&1&5 \\
\end{tabular}
\end{minipage}
\end{center}
Note that $(G,*)$ is isomorphic to $C_6$, the cyclic group of order $6$, and $(G,\cdot)= L$ is a nonassociative loop. Consider the mapping $f: C_6\rightarrow L$ defined by $f(x) = x$, for all $x\in G$. For $x,y\in G$ such that $x\leq y$ and $(x,y)\not = (3,5)$, we have $y*x = x*y = x\cdot y$. Furthermore, $3*5 = 5*3 = 5\cdot 3$. Thus, $f$ is a half-isomorphism. From $3\cdot 5 = 4$ and $3*5 = 5*3 = 1$, it follows that $f^{-1}(3\cdot 5)\not \in \{f^{-1}(3)*f^{-1}(5),f^{-1}(5)*f^{-1}(3)\}$, and hence $f^{-1}$ is not a half-isomorphism.
\qed
\end{exem}
We note that providing some examples for the case of non-special half-automorphisms can be very complicated. For finite loops, every half-automorphism is special \cite[Corollary 2.7]{GA192}, and in section~\ref{sec3} we show that the same is valid for finite groupoids.
As we can see in the example~\ref{ex1}, in general, a half-isomorphism does not preserve the structure of the loop. For instance, $C_6$ is associative and commutative and has a subgroup $H = \{1,3,5\}$, while $L$ is nonassociative and noncommutative, and $f(H)$ is not a subloop of $L$. However, the inverse mapping of a half-isomorphism can preserve some structure, like the commutative property and subloops \cite[Proposition 2.2]{GA192}. The same naturally holds for special half-isomorphisms.
This paper is organized as follows: Section~\ref{sec2} presents the definitions and basic results about half-isomorphisms. In section~\ref{sec3}, some presented results in \cite{GA192} on half-isomorphisms in loops are generalized to groupoids. In section~\ref{sec4}, the concept of \emph{principal h-groupoid} of a groupoid is defined, and then a necessary and sufficient condition for the existence of special half-isomorphisms between groupoids is obtained. Furthermore, equations related to the number of special half-automorphisms, automorphisms and anti-automorphisms of a groupoid are obtained. In section~\ref{sec5}, the concept of \emph{principal h-quasigroup} of a quasigroup is defined, and then the set of these quasigroups is described. Some examples of non-special half-automorphisms in loops are provided in section~\ref{sl}.
\section{Preliminaries}
\label{sec2}
Here, the required definitions and basic results on half-isomorphisms are stated.
\begin{defi} Let $G$ and $G'$ be groupoids. We will say that $G$ is \emph{half-isomorphic} to $G'$, denoted by $G\stackrel{H}{\cong} G'$, if there exists a special half-isomorphism between $G$ and $G'$. Note that $\stackrel{H}{\cong}$ is an equivalence relation. If $G$ is isomorphic to $G'$, we write $G \cong G'$.
\end{defi}
The next proposition assures that quasigroups half-isomorphic to loops are also loops.
\begin{prop}
\label{prop21}
Let $(G,*)$ and $(G',\cdot)$ be groupoids and $f:G \rightarrow G'$ be a half-isomorphism. If $G'$ has an identity element $1$, then $f^{-1}(1)$ is the identity element of $G$.
\end{prop}
\begin{proof}
Let $x = f^{-1}(1) \in G$. For $y\in G$, we have that $\{f(x*y),f(y*x)\}\subset \{1\cdot f(y),f(y)\cdot1\} = \{f(y)\}$. Since $f$ is a bijection, we have $x*y = y*x = y$. Therefore, $x$ is an identity element of $G$.
\end{proof}
Now, let $(G,*),(G',\cdot),(G'',\bullet)$ be groupoids, and $f:G \rightarrow G'$ and $g:G' \rightarrow G''$ be half-isomorphisms. For $x,y\in G$, we have
\begin{center}
$gf(x*y) \in \{g(f(x)\cdot f(y)),g(f(y)\cdot f(x))\} = \{gf(x)\bullet gf(y),gf(y)\bullet gf(x))\}.$
\end{center}
Thus, $gf$ is a half-isomorphism. If $f$ and $g$ are special half-isomorphisms, then $(gf)^{-1} = f^{-1}g^{-1}$ is also a special half-isomorphism.
We denote the sets of the half-automorphisms, special half-automorphisms, and trivial half-automorphisms of a groupoid $G$ by $\mathit{Half}(G)$, $\mathit{Half}_S(G)$, and $\mathit{Half_T}(G)$, respectively. Note that automorphisms and anti-automorphisms are always special half-automorphisms, and consequently $\mathit{Half_T}(G) \subset \mathit{Half}_S(G) \subset \mathit{Half}(G)$.
For $f,g\in \mathit{Half}(G)$, we already see that $fg\in \mathit{Half}(G)$. The identity mapping $I_d$ of $G$ is the identity element of $\mathit{Half}(G)$. Thus, $\mathit{Half}(G)$ is a group if and only if it is closed under inverses, which is equivalent to $\mathit{Half}(G) = \mathit{Half}_S(G)$. In particular, $\mathit{Half}_S(G)$ is always a group.
A composition of two automorphisms or two anti-automorphisms is an automorphism, and if $f$ is an automorphism and $g$ is an anti-automorphism, then $fg$ and $gf$ are anti-automorphisms and $g^{-1}fg$ is an automorphism. Thus, $\mathit{Half_T}(G)$ is a group and the automorphism group of $G$, denoted by $Aut(G)$, is a normal subgroup of $\mathit{Half_T}(G)$.
The following result summarizes the discussion above.
\begin{prop}
\label{prop22}
Let $G$ be a groupoid. Then:
\noindent{}(a) $\mathit{Half}_S(G)$ is a group and $\mathit{Half}_T(G)$ is a subgroup of $\mathit{Half}_S(G)$.
\\
(b) $\mathit{Half}(G)$ is a group if and only if $\mathit{Half}(G)=\mathit{Half}_S(G)$.
\\
(c) $Aut(G) \triangleleft \mathit{Half}_T(G)$.
\end{prop}
\begin{obs}
It is shown in section~\ref{sl} that in general $\mathit{Half}(G)$ is not a group.
\end{obs} | 3,131 | 14,878 | en |
train | 0.31.1 | \section{Preliminaries}
\label{sec2}
Here, the required definitions and basic results on half-isomorphisms are stated.
\begin{defi} Let $G$ and $G'$ be groupoids. We will say that $G$ is \emph{half-isomorphic} to $G'$, denoted by $G\stackrel{H}{\cong} G'$, if there exists a special half-isomorphism between $G$ and $G'$. Note that $\stackrel{H}{\cong}$ is an equivalence relation. If $G$ is isomorphic to $G'$, we write $G \cong G'$.
\end{defi}
The next proposition assures that quasigroups half-isomorphic to loops are also loops.
\begin{prop}
\label{prop21}
Let $(G,*)$ and $(G',\cdot)$ be groupoids and $f:G \rightarrow G'$ be a half-isomorphism. If $G'$ has an identity element $1$, then $f^{-1}(1)$ is the identity element of $G$.
\end{prop}
\begin{proof}
Let $x = f^{-1}(1) \in G$. For $y\in G$, we have that $\{f(x*y),f(y*x)\}\subset \{1\cdot f(y),f(y)\cdot1\} = \{f(y)\}$. Since $f$ is a bijection, we have $x*y = y*x = y$. Therefore, $x$ is an identity element of $G$.
\end{proof}
Now, let $(G,*),(G',\cdot),(G'',\bullet)$ be groupoids, and $f:G \rightarrow G'$ and $g:G' \rightarrow G''$ be half-isomorphisms. For $x,y\in G$, we have
\begin{center}
$gf(x*y) \in \{g(f(x)\cdot f(y)),g(f(y)\cdot f(x))\} = \{gf(x)\bullet gf(y),gf(y)\bullet gf(x))\}.$
\end{center}
Thus, $gf$ is a half-isomorphism. If $f$ and $g$ are special half-isomorphisms, then $(gf)^{-1} = f^{-1}g^{-1}$ is also a special half-isomorphism.
We denote the sets of the half-automorphisms, special half-automorphisms, and trivial half-automorphisms of a groupoid $G$ by $\mathit{Half}(G)$, $\mathit{Half}_S(G)$, and $\mathit{Half_T}(G)$, respectively. Note that automorphisms and anti-automorphisms are always special half-automorphisms, and consequently $\mathit{Half_T}(G) \subset \mathit{Half}_S(G) \subset \mathit{Half}(G)$.
For $f,g\in \mathit{Half}(G)$, we already see that $fg\in \mathit{Half}(G)$. The identity mapping $I_d$ of $G$ is the identity element of $\mathit{Half}(G)$. Thus, $\mathit{Half}(G)$ is a group if and only if it is closed under inverses, which is equivalent to $\mathit{Half}(G) = \mathit{Half}_S(G)$. In particular, $\mathit{Half}_S(G)$ is always a group.
A composition of two automorphisms or two anti-automorphisms is an automorphism, and if $f$ is an automorphism and $g$ is an anti-automorphism, then $fg$ and $gf$ are anti-automorphisms and $g^{-1}fg$ is an automorphism. Thus, $\mathit{Half_T}(G)$ is a group and the automorphism group of $G$, denoted by $Aut(G)$, is a normal subgroup of $\mathit{Half_T}(G)$.
The following result summarizes the discussion above.
\begin{prop}
\label{prop22}
Let $G$ be a groupoid. Then:
\noindent{}(a) $\mathit{Half}_S(G)$ is a group and $\mathit{Half}_T(G)$ is a subgroup of $\mathit{Half}_S(G)$.
\\
(b) $\mathit{Half}(G)$ is a group if and only if $\mathit{Half}(G)=\mathit{Half}_S(G)$.
\\
(c) $Aut(G) \triangleleft \mathit{Half}_T(G)$.
\end{prop}
\begin{obs}
It is shown in section~\ref{sl} that in general $\mathit{Half}(G)$ is not a group.
\end{obs}
\section{Special half-isomorphisms on groupoids}
\label{sec3}
Considering $(G,*)$ and $(G',\cdot)$ as groupoids, define the following set:
\begin{equation*}
K(G) = \{(x,y)\in G\times G \mid xy = yx\}
\end{equation*}
The next two results are respectively extensions of Proposition 2.3 and Theorem 2.5 of \cite{GA192} to groupoids. We note that the proofs are similar to the ones for corresponding results given in \cite{GA192}.
\begin{lema}
\label{lema32}
Let $f:G \rightarrow G'$ be a half-isomorphism. Then
\begin{equation*}
\begin{aligned}
\psi_{(G,G')}:{} & K(G') &\rightarrow {}&K(G)\\
&(x,y)&\mapsto {}&(f^{-1}(x),f^{-1}(y))
\end{aligned}
\end{equation*}
\noindent{}is injective.
\end{lema}
\begin{proof}
For $(x,y)\in K(G')$, we have
\begin{equation*}
\{f(f^{-1}(x)*f^{-1}(y)),f(f^{-1}(y)*f^{-1}(x))\} \subseteq \{x\cdot y, y\cdot x\} = \{x\cdot y\}.
\end{equation*}
Then, $f(f^{-1}(x)*f^{-1}(y))=f(f^{-1}(y)*f^{-1}(x))$, and so $f^{-1}(x)*f^{-1}(y) = f^{-1}(y)*f^{-1}(x)$. Thus, $(f^{-1}(x),f^{-1}(y))\in K(Q)$ and the mapping $\psi_{(G,G')}$ is well-defined.
Now, let $(x,y),(x',y')\in K(G')$ such that $\psi_{(G,G')}((x,y)) = \psi_{(G,G')}((x',y'))$. Then, $f^{-1}(x) = f^{-1}(x')$ and $f^{-1}(y) = f^{-1}(y')$. Since $f$ is a bijection, the mapping $\psi_{(G,G')}$ is injective.
\end{proof}
\begin{teo}
\label{teo31}
Let $f:G \rightarrow G'$ be a half-isomorphism. Then, the following statements are equivalent:
\noindent{}(a) $f$ is special.\\
(b) $\{f(x*y),f(y*x)\} = \{f(x)\cdot f(y), f(y)\cdot f(x)\}$ for any $x,y\in G$.\\
(c) For all $x,y\in G$ such that $x*y = y*x$, we have $f(x)\cdot f(y) = f(y)\cdot f(x)$.\\
(d) $\psi_{(G,G')}$ is a bijection.
\end{teo}
\begin{proof}(a) $\Rightarrow$ (b) Let $x,y \in G$. Since $f$ is a half-isomorphism, we have \mbox{$\{f(x*y),f(y*x)\} \subseteq$} \mbox{$ \{f(x)\cdot f(y), f(y)\cdot f(x)\}$.} Since $f^{-1}$ is a half-isomorphism, we have $\{f^{-1}(f(x)\cdot f(y)), f^{-1}(f(y)\cdot f(x))\} \subseteq \{x*y,y*x\}$, and hence $\{f(x)\cdot f(y), f(y)\cdot f(x)\} \subseteq \{f(x*y),f(y*x)\}$.\\
\noindent{}(b) $\Rightarrow$ (c) Let $x,y \in G$ such that $x*y = y*x$. Then, $f(x*y) = f(y*x)$. Using the hypothesis, we get $\{f(x)\cdot f(y), f(y)\cdot f(x)\} = \{f(x*y),f(y*x)\} = \{f(x*y)\}$, and therefore $f(x)\cdot f(y) = f(y)\cdot f(x)$.\\
\noindent{}(c) $\Rightarrow$ (d) From Lemma~\ref{lema32}, we know that $\psi_{(G,G')}$ is injective. Let $(x,y) \in K(G)$. By hypothesis, we have $f(x)\cdot f(y) = f(y)\cdot f(x)$, and then $(f(x),f(y)) \in K(G')$. It is clear that $\psi_{(G,G')}((f(x),f(y))) = (x,y)$, and hence $\psi_{(G,G')}$ is a bijection.\\
\noindent{}(d) $\Rightarrow$ (a) Let $x,y\in G'$. If $(x,y)\in K(G')$, then $(f^{-1}(x),f^{-1}(y)) \in K(G)$ since $\psi_{(G,G')}$ is a bijection. Thus, $f(f^{-1}(x)* f^{-1}(y)) = x\cdot y$, and therefore $f^{-1}(x\cdot y) = f^{-1}(x)* f^{-1}(y)$. If $(x,y)\not \in K(G')$, then $(f^{-1}(x),f^{-1}(y)) \not \in K(G)$ since $\psi_{(G,G')}$ is a bijection. Consequently, we have
\begin{center}
$\{f(f^{-1}(x)* f^{-1}(y)), f(f^{-1}(y)* f^{-1}(x))\} = \{x\cdot y,y\cdot x\}$,
\end{center}
\noindent{}and hence $f^{-1}(x\cdot y) \in \{f^{-1}(x)* f^{-1}(y), f^{-1}(y)* f^{-1}(x)\}$.
\end{proof}
As direct consequences of Lemma~\ref{lema32} and Theorem~\ref{teo31}, we have the following corollaries.
\begin{cor}
\label{cor31}
Let $f:G \rightarrow G'$ be a half-isomorphism. If $|K(G)| = |K(G')| < \infty$, then $f$ is special.
\end{cor}
\begin{cor}
\label{cor32}
Let $G$ be a groupoid such that $|K(G)| < \infty$. Then, $\mathit{Half}(G)$ is a group.
\end{cor}
\begin{cor}
\label{cor33}
Let $G$ be a finite groupoid. Then, $\mathit{Half}(G)$ is a group.
\end{cor}
A loop is \emph{diassociative} if any two of its elements generate an associative subloop. Moufang loops and groups are examples of diassociative loops. In \cite[Lemma 2.1]{KSV16}, the authors showed that the item (c) of Theorem~\ref{teo31} holds for any half-isomorphism on diassociative loops. Therefore, we have the next result.
\begin{cor}
\label{cor34}
Let $(L,*)$ and $(L',\cdot)$ be diassociative loops. Then, every half-isomorphism between $L$ and $L'$ is special.
\end{cor}
\begin{obs} The Corollary~\ref{cor34} cannot be extended for some important classes of loops. In example~\ref{ex61}, a non-special half-isomorphism between a right Bol loop and a group is introduced. A loop is called \emph{right Bol loop} if it satisfies the identity $((xy)z)y = x((yz)y)$.
\end{obs}
This section is finished with a property of half-isomorphic groupoids.
\begin{prop}
\label{prop33}
If $G\stackrel{H}{\cong} G'$, then:
\noindent{}(a) $\mathit{Half}(G) \cong \mathit{Half}(G')$
\\
(b) $\mathit{Half}_S(G) \cong \mathit{Half}_S(G')$
\end{prop}
\begin{proof}
Let $\phi: G \rightarrow G'$ be a special half-isomorphism. Define $\varphi: \mathit{Half}(G) \rightarrow \mathit{Half}(G')$ by $\varphi(f) = \phi f \phi^{-1}$. It is clear that $\varphi$ is a bijection. For $f,g\in \mathit{Half}(G)$, we have $\varphi(fg) = \phi fg \phi^{-1} = \phi f\phi^{-1} \phi g \phi^{-1} = \varphi(f)\varphi(g)$. Thus, $\mathit{Half}(G) \cong \mathit{Half}(G')$. The rest of the claim is concluded from the fact that $\varphi(\mathit{Half}_S(G)) = \mathit{Half}_S(G')$.
\end{proof}
\begin{obs}
\label{ob31} If $G\stackrel{H}{\cong} G'$, then $Aut(G)$ is not isomorphic to $Aut(G')$ in general (see example~\ref{ex41}).
\end{obs} | 3,447 | 14,878 | en |
train | 0.31.2 | \section{Principal h-groupoids of G}
\label{sec4}
In this section, $G_0 = (G,*)$ is considered as a noncommutative groupoid.
Let $(G',\bullet)$ be a groupoid such that $G_0\stackrel{H}{\cong} (G',\bullet)$. Then, there exists a special half-isomorphism $f$ of $G_0$ into $(G',\bullet)$. Define an operation $\cdot$ on $G$ by $x\cdot y = f^{-1}(f(x)\bullet f(y))$. Thus, $f$ is an isomorphism of $(G,\cdot)$ into $(G',\bullet)$, and hence $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism, where $I_d$ is the identity mapping of $G$.
A groupoid $(G,\cdot)$ for which $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism is called a \emph{principal h-groupoid of $G_0$}. Therefore, the following result is at hand.
\begin{prop}
\label{prop41}
Let $G'$ be a groupoid. Then, $G_0\stackrel{H}{\cong} G'$ if and only if $G'$ is isomorphic to a principal h-groupoid of $G_0$.
\end{prop}
Denote by $\mathcal{M}(G_0)$ the set of the principal h-groupoids of $G_0$. Note that for \mbox{$(G,\cdot),(G,\bullet)\in \mathcal{M}(G_0)$,} we have $(G,\cdot)=(G,\bullet)$ if $x\cdot y = x \bullet y$, for all $x,y\in G$, which is equivalent to $I_d$ being an isomorphism between $(G,\cdot)$ and $(G,\bullet)$.
Let $(G,\cdot)\in \mathcal{M}(G_0)$. Since $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism, we have
\begin{equation}
\label{eq41}
\{x*y,y*x\} = \{x\cdot y,y\cdot x\}, \textrm{ for all } x,y \in G.
\end{equation}
If $(x,y)\in K(G_0)$, then $x\cdot y = y\cdot x = x*y$. For each pair $(x,y),(y,x)\in G\times G\setminus K(G_0)$, there are two possible values for $x\cdot y$ and $y\cdot x$ by \eqref{eq41}. Thus, if $G$ is finite, we have $2^{| G\times G\setminus K(G_0)|/2}$ possibilities for a principal h-groupoid of $G_0$. Hence, the following result is at hand.
\begin{prop}
\label{prop42} If $G$ is finite, then $|\mathcal{M}(G_0)| =2^{(|G|^2-|K(G_0)|)/2}$.
\end{prop}
Define $\mathcal{M}_I(G_0) = \{G'\in \mathcal{M}(G_0)\,|\, G'\cong G_0\}$ and let $S(G)$ be the set of permutations of $G$. For $G' = (G,\cdot) \in \mathcal{M}_I(G)$, define $Iso(G',G_0) = \{f\in S(G)\,|\, f \textrm{ is an isomorphism of } G' \textrm{ into } G_0\}$. Note that $Iso(G_0,G_0) = Aut(G_0)$. In the next result, we determine a relationship between $\mathit{Half}_S(G_0)$, $Aut(G_0)$ and $\mathcal{M}_I(G_0)$.
\begin{prop}
\label{prop43} We have:
\noindent{}(a) $Iso(G',G_0) \subset \mathit{Half}_S(G_0)$, for every $G'\in \mathcal{M}_I(G_0)$.
\\
(b) For each $G'\in \mathcal{M}_I(G_0)$, $Iso(G',G_0)$ is a right coset of $Aut(G_0)$ in $\mathit{Half}_S(G_0)$, that is, there exists $f\in \mathit{Half}_S(G_0)$ such that $Iso(G',G_0) = Aut(G_0)f$.
\\
(c) For $G_1,G_2 \in \mathcal{M}_I(G_0)$, if $Iso(G_1,G_0) \cap Iso(G_2,G_0) \not = \emptyset$, then $G_1 = G_2$.
\\
(d) $\mathit{Half}_S(G_0) = \bigcup_{G'\in \mathcal{M}_I(G_0)} Iso(G',G_0)$.
\\
(e) $|\mathcal{M}_I(G_0)| = [\mathit{Half}_S(G_0):Aut(G_0)]$, which is the index of $Aut(G_0)$ in $\mathit{Half}_S(G_0)$.
\end{prop}
\begin{proof} (a) For $G' = (G,\cdot)\in \mathcal{M}_I(G_0)$, let $f\in Iso(G',G_0)$. Then $f(x\cdot y) = f(x)*f(y)$, for all $x,y\in G$. By \eqref{eq41}, $\{f(x\cdot y),f(y\cdot x)\} = \{f(x)*f(y),f(y)*f(x)\}$, for all $x,y \in G$. By Theorem~\ref{teo31}, $f\in \mathit{Half}_S(G_0)$.\\
\noindent{}(b) Fix $f\in Iso(G',G_0)$. It is clear that $gf^{-1}\in Aut(G_0)$, for every $g\in Iso(G',G_0)$, and $\alpha f \in Iso(G',G_0)$, for every $\alpha \in Aut(G_0)$. Hence, we have the desired result.\\
\noindent{}(c) Let $f\in Iso(G_1,G_0) \cap Iso(G_2,G_0)$. Note that $I_d= f^{-1}f:G_1 \rightarrow G_2$ is an isomorphism. From the definition of $\mathcal{M}(G_0)$, it follows that $G_1 = G_2$.\\
\noindent{}(d) Let $f \in \mathit{Half}_S(G_0)$. Define the operation $\cdot$ on $G$ by $x\cdot y = f^{-1}(f(x)*f(y))$, for all $x,y\in G$. Note that $f: (G,\cdot)\rightarrow (G,*)$ is an isomorphism. Furthermore, since $f \in \mathit{Half}_S(G_0)$, and $f(x\cdot y) = f(x)*f(y)$ and $f(y\cdot x) = f(y)*f(x)$, for all $x,y \in G$, we have $\{x\cdot y,y\cdot x\} = \{x*y,y*x\}$, for all $x,y \in G$. Thus, $G' = (G,\cdot)\in \mathcal{M}_I(G_0)$, and hence $f\in Iso(G',G_0)$.\\
\noindent{}(e) It is a consequence of the previous items.
\end{proof}
As a consequence of the Proposition~\ref{prop33} and the item (e) of Proposition~\ref{prop43}, we have the following result.
\begin{cor}
\label{cor41} Let $G',G''$ be groupoids such that $G'\stackrel{H}{\cong} G''$ and $\mathit{Half}_S(G')$ is finite. Then,
\begin{center}
$|\mathcal{M}_I(G')|.|Aut(G')| = |\mathcal{M}_I(G'')|.|Aut(G'')|$
\end{center}
\end{cor}
Define $G_0^T = (G,\cdot)$, where $x\cdot y = y*x$, for all $x,y\in G$, and denote the set of anti-automorphisms of $G_0$ by $Ant(G_0)$. Since $G_0$ is noncommutative, we have $Aut(G_0)\cap Ant(G_0) = \emptyset$.
\begin{prop}
\label{prop44} $G_0$ has an anti-automorphism if and only if $G_0^T\in \mathcal{M}_I(G_0)$. In this case, $|Ant(G)| = |Aut(G)|$.
\end{prop}
\begin{proof} Note that a bijection $f$ of $G$ is an anti-automorphism of $G_0$ if and only if $f$ is an isomorphism of $G_0$ into $G_0^T$. The rest of the claim is concluded from the item (b) of Proposition~\ref{prop43}.
\end{proof}
\begin{exem}
\label{ex41}
Let $Q = \{1,2,...,8\}$ and consider the following Cayley tables of $(Q,*)$ and $(Q,\cdot)$:
\begin{center}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$*$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&6&5&7&8\\
2&2&1&4&3&5&6&8&7\\
3&4&3&1&2&7&8&5&6\\
4&3&4&2&1&8&7&6&5\\
5&5&6&8&7&1&2&4&3\\
6&6&5&7&8&2&1&3&4\\
7&8&7&6&5&3&4&1&2\\
8&7&8&5&6&4&3&2&1\\
\end{tabular}
\end{minipage}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$\cdot $&1&2&3&4&5&6&7&8\\
\hline
1&1&2&4&3&6&5&7&8\\
2&2&1&3&4&5&6&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&1&8&7&6&5\\
5&5&6&8&7&1&2&4&3\\
6&6&5&7&8&2&1&3&4\\
7&8&7&6&5&3&4&1&2\\
8&7&8&5&6&4&3&2&1\\
\end{tabular}
\end{minipage}
\end{center}
We have $(Q,*)$ and $(Q,\cdot)$ being quasigroups. Note that, for $x,y\in Q$:
\begin{center}
$x*y = \left\{\begin{array}{l}
y\cdot x, \textrm{ if } (x,y) \in \{(1,3),(1,4),(2,3),(2,4),(3,1),(3,2),(4,1),(4,2)\},\\
x\cdot y, \textrm{ otherwise.}
\end{array}\right.$
\end{center}
Thus, $(Q,\cdot) \in \mathcal{M}((Q,*))$. Using the LOOPS package \cite{NV1} for GAP \cite{gap} we get $|Aut((Q,*))| = 4$ and $|Aut((Q,\cdot))| = 8$. This illustrates Remark~\ref{ob31}.
Note that $|K((Q,*))| = 16$, and hence $|\mathcal{M}((Q,*))| = 2^{24} = 16777216$. Using a GAP computation with the LOOPS package, we get that there are $64$ quasigroups in $\mathcal{M}((Q,*))$ and $|\mathcal{M}_I((Q,*))| = 12$. By Proposition~\ref{prop43}, we have $|\mathit{Half}((Q,*))|= 48$ and $|\mathcal{M}_I((Q,\cdot))| = 6$.
It is observed that the number of quasigroups in $\mathcal{M}((Q,*))$ is much smaller than $|\mathcal{M}((Q,*))|$. In the next section, we will see that the same occurs for any finite noncommutative quasigroup. \qed
\end{exem} | 3,213 | 14,878 | en |
train | 0.31.3 | \section{Principal h-quasigroups of Q}
\label{sec5}
Here, $Q_0= (Q,*)$ is considered as a noncommutative quasigroup. A quasigroup $(Q,\cdot)$ is a \emph{principal h-quasigroup of $Q_0$} if $(Q,\cdot)\in \mathcal{M}(Q_0)$. Denote by $\mathcal{N}(Q_0)$ the set of the principal h-quasigroups of $Q_0$. It is clear that $\mathcal{M}_I(Q_0) \subset \mathcal{N}(Q_0) \subset \mathcal{M}(Q_0)$. The next result is concluded from Proposition~\ref{prop41}.
\begin{prop}
\label{prop51} Let $Q'$ be a quasigroup. Then $Q_0\stackrel{H}{\cong} Q'$ if and only if $Q'$ is isomorphic to a principal h-quasigroup of $Q_0$.
\end{prop}
Now, we describe $\mathcal{N}(Q_0)$. For $(x,y),(x',y')\in Q\times Q \setminus K(Q_0)$, we say that $(x,y)\sim (x',y')$ if one of the following holds:
\noindent{}(i) $(x',y') = (y,x)$,
\\
(ii) $x = x'$ and $\{x*y,y*x\}\cap \{x*y',y'*x\}\not = \emptyset$,
\\
(iii) $y = y'$ and $\{x*y,y*x\}\cap \{x'*y,y*x'\}\not = \emptyset$.
We say that $(x,y)\equiv (x',y')$ if there are $z_1,z_2,...,z_l \in Q\times Q \setminus K(Q_0)$ such that $(x,y)\sim z_1 \sim z_2\sim...\sim z_l \sim (x',y')$.
The relation $\sim$ is reflexive and symmetric, and hence $\equiv$ is an equivalence relation. Denote by $r(Q_0)$ the number of equivalence classes of $\equiv$ on $Q\times Q \setminus K(Q_0)$.
Suppose that $Q$ is finite and let $\tau = \{(x_1,y_1),(x_2,y_2),...,(x_{r(Q_0)},y_{r(Q_0)})\}$ be a set of representatives of the equivalence classes of $\equiv$ on $Q\times Q \setminus K(Q_0)$. Consider $\mathbb{Z}_2 = \{0,1\}$, and for $\sigma = \{\sigma_1,\sigma_2,...,\sigma_{r(Q_0)}\} \in \mathbb{Z}_2^{r(Q_0)}$, define the operation $\stackrel{\sigma}{\bullet}$ on $Q$ by:
\begin{center}
$x \stackrel{\sigma}{\bullet} y = \left\{\begin{array}{l}
x * y, \textrm{ if } (x,y)\in K(Q_0) \textrm{ or } (x,y)\equiv (x_i,y_i), \textrm{ where } \sigma_i = 0,\\
y * x, \textrm{ if } (x,y)\equiv (x_i,y_i), \textrm{ where } \sigma_i = 1.
\end{array}\right.$
\end{center}
Denote $(Q,\stackrel{\sigma}{\bullet})$ by $Q_\sigma$ and let $\mathcal{N}_\tau(Q_0) = \{Q_\sigma \,|\, \sigma \in \mathbb{Z}_2^{r(Q_0)}\}$. Note that $\mathcal{N}_\tau(Q_0)\subset \mathcal{M}(Q_0)$ and $|\mathcal{N}_\tau(Q_0)| = 2^{r(Q_0)}$.
\begin{teo}
\label{teo51} If $Q$ is finite, then $\mathcal{N}(Q_0) = \mathcal{N}_\tau(Q_0)$. In particular, $|\mathcal{N}(Q_0)| = 2^{r(Q_0)}$.
\end{teo}
\begin{proof}
Let $Q_\sigma \in \mathcal{N}_\tau(Q_0)$. Since $Q$ is finite, in order to prove that $Q_\sigma$ is a quasigroup, we only need to show that the cancellation laws are satisfied, that is, $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y' \Rightarrow y = y'$ and $x \stackrel{\sigma}{\bullet} y = x' \stackrel{\sigma}{\bullet} y \Rightarrow x = x'$.
Let $x,y,y'\in Q$ be such that $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y'$. If $(x,y)\in K(Q_0)$, then \mbox{$x*y = y*x \in $} $ \{x*y',y'*x\}$, and hence $y = y'$. Now suppose that $(x,y)\not \in K(Q_0)$. We have four possibilities:
\noindent{}(i) $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$,
\\
(ii) $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$,
\\
(iii) $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$,
\\
(iv) $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$.
In (i) and (ii), it is immediately seen that $y = y'$.
For (iii) and (iv), we have $(x,y)\sim (x,y')$. Hence, there exists $(x_i,y_i)\in \tau$ such that $(x,y) \equiv (x_i,y_i)$ and $(x,y') \equiv (x_i,y_i)$. By definition of $\stackrel{\sigma}{\bullet}$, we have either $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$, or $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$. Since $(x,y)\not \in K(Q_0)$, it follows that $(x,y')\in K(Q_0)$. Similarly to the case $(x,y)\in K(Q_0)$, one can conclude that $y = y'$.
Thus, the cancellation law $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y' \Rightarrow y = y'$ holds in $Q_\sigma$. The second cancellation law can be proven similarly. Therefore, $Q_\sigma \in \mathcal{N}(Q_0)$.
Conversely, let $Q' = (Q,\cdot) \in \mathcal{N}(Q_0)$. Then, there exists $\sigma \in \mathbb{Z}_2^{r(Q_0)}$ such that $x_i\cdot y_i = x_i \stackrel{\sigma}{\bullet} y_i$, for any $(x_i,y_i)\in \tau$. For $(x,y)\in K(Q_0)$, it is vividly deduced that $x\cdot y = x \stackrel{\sigma}{\bullet} y$.
Consider $(x_i,y_i)\in \tau$. Then, $y_i\cdot x_i = y_i \stackrel{\sigma}{\bullet} x_i$. Let $(x,y) \in Q\times Q\setminus \{(x_i,y_i),(y_i,x_i)\}$ such that $(x,y)\sim (x_i,y_i)$. By \eqref{eq41} and the definition of $\stackrel{\sigma}{\bullet}$, we have $x\cdot y \not = x_i\cdot y_i = x_i\stackrel{\sigma}{\bullet} y_i$ and $x \stackrel{\sigma}{\bullet} y \not = x_i\stackrel{\sigma}{\bullet} y_i$, and therefore the only possibility is $x\cdot y = x \stackrel{\sigma}{\bullet} y$. For every $(x,y)\sim (x_i,y_i)$, one can use the previous arguments and result in $x'\cdot y' = x' \stackrel{\sigma}{\bullet} y'$, for all $(x',y')\sim (x,y)$. Since $Q$ is finite, this procedure must end at some point, and hence $x\cdot y = x \stackrel{\sigma}{\bullet} y$, for all $(x,y)\equiv (x_i,y_i)$. As a result, we have $Q' = Q_\sigma$.
\end{proof}
By Proposition~\ref{prop42}, if $Q$ is finite, then $r(Q_0)\leq (|Q|^2 - |K(Q_0)|)/2$. The next proposition provides a better estimate for $r(Q_0)$. According to this result, it is seen that $|\mathcal{N}(Q_0)|$ is much smaller that $|\mathcal{M}(Q_0)|$.
\begin{prop}
\label{prop52a} If $Q$ is finite, then $r(Q_0)\leq (|Q|^2 - |K(Q_0)|)/6$ and $|\mathcal{N}(Q_0)|\leq \sqrt[3]{|\mathcal{M}(Q_0)|}$. In particular, $|\mathcal{M}(Q_0)| \geq 8$.
\end{prop}
\begin{proof}
Let $(x,y)\in Q\times Q \setminus K(Q_0)$ and $[(x,y)]$ be the equivalence class of $(x,y)$ with respect to $\equiv$. Since $Q_0$ is a quasigroup, there are $x',y' \in Q$ such that $x'\not =x$, $y'\not = y$, $(x',y)\sim (x,y)$, and $(x,y')\sim (x,y)$. We have $x\not = y$, $x'\not = y$ and $x\not = y'$ since $(x,y),(x',y),(x,y')\not \in K(Q_0)$. Thus, $|[(x,y)]|\geq |\{(x,y),(x',y),(x,y'),(y,x),(y,x'),(y',x)\}| = 6$. Hence, $|Q\times Q \setminus K(Q_0)|\geq 6 \,r(Q_0)$. The rest of the claim follows from Proposition~\ref{prop42}, Theorem~\ref{teo51} and the fact that $r(Q_0)\geq 1$.
\end{proof}
If $Q$ is finite and $r(Q_0)$ is small, one can generate all quasigroups of $\mathcal{N}(Q_0)$ computationally. Then, by using Propositions~\ref{prop51} and \ref{prop44} it can be verified if a quasigroup $Q'$ is half-isomorphic to $Q_0$ and generated all elements of $\mathit{Half}(Q_0)$. However, $r(Q_0)$ can be a large number even for groups of small order, and therefore generating all the quasigroups of $\mathcal{N}(Q_0)$ becomes computationally unviable. The next example illustrates both situations. In this example, $r(Q_0)$ and $|\mathcal{M}(Q_0)|$ are obtained by using GAP computing with the LOOPS package \cite{gap,NV1}.
\begin{exem} (a) Let $A_5$ be the alternating group of order $60$. We have that $r(A_5) = 91$, and hence $|\mathcal{N}(A_5)| = 2^{91}$. Furthermore, $|\mathcal{M}(A_5)| = 2^{1650}$.
\noindent{}(b) The LOOPS package for GAP contains all nonassociative right Bol loops of order $141$ (there are $23$ such loops). The right Bol loops of this order were classified in \cite{KNV}. If $L$ is one of these loops, then $3\leq r(L) \leq 8$, and hence $|\mathcal{N}(L)| \leq 256$. Furthermore, $|\mathcal{M}(L)| \geq 2^{5405}$.\qed
\end{exem}
By Proposition~\ref{prop21}, every quasigroup half-isomorphic to a loop is also a loop. Consequently, the same results as those presented for quasigroups in this section can be proven for loops. For more structured classes of loops, as it is seen in the following result, one can provide more information about the loops of $\mathcal{N}(L)$.
\begin{prop}
\label{prop52}
Let $G$ be a finite noncommutative group. Then, $|\mathcal{M}_I(G)| = 2$.
\end{prop}
\begin{proof}
From Scott's result \cite{Sco57}, we have $\mathit{Half}(G) = \mathit{Half}_T(G)$. Since $G$ is noncommutative, the mapping $J: G \to G$, defined by $J(x) = x^{-1}$, is an anti-automorphism of $G$. By Proposition~\ref{prop44}, we have $|\mathit{Half}(G)| = 2|Aut(G)|$. Thus, the claim follows from Proposition~\ref{prop43}.
\end{proof}
In fact, the previous proposition can be extended to any noncommutative loop that has an anti-automorphism and where every half-automorphism is trivial, such as the noncommutative loops of the subclass of Moufang loops in \cite[Thereom 1.4]{KSV16}, which include the noncommutative Moufang loops of odd order \cite{GG12}. Notice that this result cannot be extended even to all Moufang loops. In \cite[Example 4.6]{G20}, a noncommutative Moufang loop $L$ of order $16$ is given for which $|\mathcal{M}_I(L)| = [\mathit{Half}(L):Aut(L)] = 16.$ | 3,515 | 14,878 | en |
train | 0.31.4 | \section{A construction of a non-special half-automorphism}
\label{sl}
Let $G$ be a nonempty set with binary operations $*$ and $\cdot$ such that there exists a non-special half-isomorphism $f:(G,*) \rightarrow (G,\cdot)$. Define $G_\infty = \prod_{i=1}^\infty G$. The elements of $G_\infty$ will be denoted by $(x_i) = (x_i)_{i=1}^\infty$, where $x_i \in G$, for all $i$. For $(x_i),(y_i)\in G_\infty$, define the operation $(x_i)\bullet (y_i) = (z_i)$, where
\begin{center}
$z_j = \left\{\begin{array}{l}
x_j*y_j, \textrm{ if } j \textrm{ is odd}, \\
x_j\cdot y_j, \textrm{ if } j \textrm{ is even.} \\
\end{array}\right.$
\end{center}
Then, $(G_\infty,\bullet)$ is a groupoid. It is easy to see that if $(G,*)$ and $(G,\cdot)$ are quasigroups (loops), then $(G_\infty,\bullet)$ is also a quasigroup (loop). Define the mapping $\phi: G_\infty\rightarrow G_\infty$ by $\phi(x_i) = (y_i)$, where
\begin{center}
$y_j = \left\{\begin{array}{l}
f(x_1), \textrm{ if } j = 2, \\
x_{j+2}, \textrm{ if } j \textrm{ is odd}, \\
x_{j-2}, \textrm{ if } j>2 \textrm{ and } j \textrm{ is even.} \\
\end{array}\right.$
\end{center}
Thus, $\phi$ is a bijection and in each entry of $(x_i)$ it behaves like an isomorphism or a half-isomorphism. Hence, $\phi$ is a half-automorphism of $G_\infty$. Since $f$ is a non-special half-isomorphism, there are $x,y \in G$ such that $f^{-1}(x\cdot y) \not \in \{f^{-1}(x)*f^{-1}(y),f^{-1}(y)*f^{-1}(x)\}$. Then,
\begin{center}
$\phi^{-1}((x)_{i=1}^\infty\bullet(y)_{i=1}^\infty) \not \in \{\phi^{-1}((x)_{i=1}^\infty)\bullet \phi^{-1}((y)_{i=1}^\infty),\phi^{-1}((y)_{i=1}^\infty)\bullet \phi^{-1}((x)_{i=1}^\infty)\}$.
\end{center}
Therefore, $\phi$ is a non-special half-automorphism of $G_\infty$.
In example~\ref{ex1}, we have loops $C_6 = (G,*)$ and $L=(G,\cdot)$ for the conditions above, hence the loop $G_\infty$ has a non-special half-automorphism. Note that $\mathit{Half}(G_\infty)$ is not a group.
In the following example, a non-special half-isomorphism between a right Bol loop and a group is provided. This example is obtained by using MACE4 \cite{mace}.
\begin{exem}
\label{ex61}
Let $G = \{1,2,...,8\}$ and consider the following Cayley tables of $(G,*)$ and $(G,\cdot)$:
\begin{center}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$*$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&6&3&5&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&8&1&7&6&5\\
5&5&6&7&1&8&2&3&4\\
6&6&5&8&7&2&1&4&3\\
7&7&8&5&3&6&4&1&2\\
8&8&7&6&5&4&3&2&1\\
\end{tabular}
\end{minipage}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$\cdot$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&3&6&5&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&1&8&7&6&5\\
5&5&7&6&8&1&3&2&4\\
6&6&8&5&7&2&4&1&3\\
7&7&5&8&6&3&1&4&2\\
8&8&6&7&5&4&2&3&1\\
\end{tabular}
\end{minipage}
\end{center}
We have $(G,*) = L$ as a right Bol loop and $(G,\cdot)$ being isomorphic to $D_8$, which is the dihedral group of order $8$. The permutation $f = (3 \, 5 \, 7)(4\, 6\, 8)$ of $G$ is a half-isomorphism of $L$ into $D_8$. Since $|K(L)| = 56$ and $|K(D_8)| = 40$, $f$ is a non-special half-isomorphism by Theorem~\ref{teo31}. Since $L$ and $D_8$ are right Bol loops, $G_\infty$ is also a right Bol loop, and from the previous construction we have a non-special half-automorphism in a right Bol loop of infinite order.\qed
\end{exem}
\section*{Acknowledgments}
Some calculations in this work have been made by using the finite model builder MACE4, developed by McCune \cite{mace}, and the LOOPS package \cite{NV1} for GAP \cite{gap}.
\addcontentsline{toc}{section}{Acknowledgments}
\end{document} | 1,572 | 14,878 | en |
train | 0.32.0 | \begin{document}
\begin{frontmatter}
\title{Bayesian estimation for a parametric Markov Renewal model applied to seismic data}
\runtitle{Bayesian Markov Renewal model for seismic data}
\author{\fnms{Ilenia} \snm{Epifani}\corref{}\ead[label=e1]{[email protected]}}
\address{Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy\\ \printead{e1}}
\and
\author{\fnms{Lucia} \snm{Ladelli}\ead[label=e2]{[email protected]}}
\address{Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy\\ \printead{e2}}
\and
\author{\fnms{Antonio} \snm{Pievatolo}\ead[label=e3]{[email protected]}}
\address{IMATI-CNR, Via Bassini 15, 20133 Milano, Italy\\ \printead{e3}}
\runauthor{Epifani, Ladelli, Pievatolo}
\begin{abstract}
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning data'' and moment existence conditions. This framework is applied to data of earthquakes of three types of severity (low, medium and high size) that occurred in the central Northern Apennines in Italy and collected by the \cite{CPTI04} catalogue. Assumptions on two types of energy accumulation and release mechanisms are evaluated.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60K20}
\kwd{62F15}
\kwd{62M05}
\kwd{86A15}
\kwd[; secondary ]{65C05}
\end{keyword}
\begin{keyword}
\kwd{Bayesian inference}
\kwd{Earthquakes}
\kwd{Gibbs sampling}
\kwd{Markov Renewal process}
\kwd{Predictive distribution}
\kwd{semi-Markov process}
\kwd{Weibull distribution}
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:introduction}
Markov Renewal processes or their semi-Markov representation have been considered in the seismological literature as models which allow the distribution of the inter-occurrence times between earthquakes to depend on the last and the next earthquake and to be not necessarily exponential.
The time predictable and the slip predictable models studied in \cite{ShimazakiNakata}, \cite{GuagMol}, \cite{GuagMolMul} and \cite{BetroGaravGuagRotonTagl} are special cases of Markov Renewal processes. These models are capable of interpreting the predictable behavior of strong earthquakes in some seismogenic areas. In these processes the magnitude is a deterministic function of the inter-occurrence time.
A stationary Markov Renewal process with Weibull inter-occurrence times has been studied from a classical statistical point of view in \cite{Alvarez}. The Weibull model allows for the consideration of monotonic hazard rates; it contains the exponential model as a special case which gives a Markov Poisson point process. In \cite{Alvarez} the model parameters were fitted to the large earthquakes in the North Anatolian Fault Zone through maximum likelihood and the Markov Poisson point process assumption was tested.
In order to capture a non monotonic behavior in the hazard, in \cite{GaravagliaPavani} the model of Alvarez was modified and a Markov Renewal process with inter-occurrence times that are mixtures of an exponential and a Weibull distribution was fitted to the same Turkish data.
In \cite{Masala} a parametric semi-Markov model with a generalized Weibull distribution for the inter-occurrence times was adapted to Italian earthquakes. Actually the semi-Markov model with generalized Weibull distributed times was first used in \cite{Foucher} to study the evolution of HIV infected patients. \cite{Votsi} considered a semi-Markov model for the seismic hazard assessment in the Northern Aegean sea and estimated the quantities of interest (semi-Markov kernel, Markov Renewal functions, etc.) through a nonparametric method.
While a wide literature concerning classical inference for Markov Renewal models for earthquake forecasting exists, to our knowledge a Bayesian approach is limited in this context. \cite{PatwardhanEtAlii} considered a semi-Markov model with log-normal distributed discrete inter-occurrence times and applied it to the large earthquakes in the circum-Pacific belt.
They stressed the fact that it is relevant to use Bayesian techniques when prior knowledge is available and it is fruitful even if the sample size is small.
\cite{MarinPlaRiosInsua} also employed semi-Markov models in the Bayesian framework, applied to a completely different area: sow farm management. They used WinBugs to perform computations (but without giving details) and they elicited their prior distributions on parameters from knowledge on farming practices.
From a probabilistic viewpoint, a Bayesian statistical treatment of a semi-Markov process amounts to model the data as a mixture of semi-Markov processes, where the mixing measure is supported on the parameters, by means of their prior laws. A complete characterization of such a mixture has been given in
\cite{EpifaniFortiniLadelli}.
In this paper we develop a parametric Bayesian analysis for a Markov Renewal process modelling earthquakes in an Italian seismic region. The magnitudes are classified into three categories according to their severity: low, medium and high size, and these categories represent the states visited by the process. As in \cite{Alvarez}, the inter-occurrence times are assumed to be Weibull random variables. The ``current sample'' is formed by the sequences of earthquakes in a homogeneous seismic region and by the corresponding inter-occurrence times collected up to a time $T$. When $T$ does not coincide with an earthquake, the last observed inter-occurrence time is censored.
The prior distribution of the parameters of the model is elicited using a ``learning dataset'', i.e. data coming from a seismic region similar to that under analysis.
The posterior distribution of the parameters is obtained through Gibbs sampling and the following summaries are estimated: transition probabilities, shape and scale parameters of the Weibull holding times for each transition and the so-called cross state-probabilities (CSPs). The transition probabilities indicate whether the strength of the next earthquake is in some way dependent on the strength of the last one; the shape parameters of the holding times indicate whether the hazard rate between two earthquakes of given magnitude classes is decreasing or increasing; the CSPs give the probability that the next earthquake occurs at or before a given time and is of a given magnitude, conditionally on the time elapsed since the last earthquake and on its magnitude.
The paper is organized as follows. In Section~\ref{sec:1} we illustrate the dataset and we discuss the choice of the Weibull model in detail. Section~\ref{sec:2} introduces the parametric Markov Renewal model.
Section~\ref{sec:3} deals with the elicitation of the prior.
Section~\ref{sec:apennines} contains the Bayesian data analysis with the estimation of the above-mentioned summaries. We also test a time predictable and a slip predictable model against the data. Section~\ref{sec:6} is devoted to some concluding remarks. Appendix \ref{sec:gibbsampling} contains the detailed derivation of the full conditional distributions and the JAGS (Just Another Gibbs Sampler) implementation of the Gibbs sampler (\cite{Plummer}). | 2,068 | 34,990 | en |
train | 0.32.1 | \section{Introduction}
\label{sec:introduction}
Markov Renewal processes or their semi-Markov representation have been considered in the seismological literature as models which allow the distribution of the inter-occurrence times between earthquakes to depend on the last and the next earthquake and to be not necessarily exponential.
The time predictable and the slip predictable models studied in \cite{ShimazakiNakata}, \cite{GuagMol}, \cite{GuagMolMul} and \cite{BetroGaravGuagRotonTagl} are special cases of Markov Renewal processes. These models are capable of interpreting the predictable behavior of strong earthquakes in some seismogenic areas. In these processes the magnitude is a deterministic function of the inter-occurrence time.
A stationary Markov Renewal process with Weibull inter-occurrence times has been studied from a classical statistical point of view in \cite{Alvarez}. The Weibull model allows for the consideration of monotonic hazard rates; it contains the exponential model as a special case which gives a Markov Poisson point process. In \cite{Alvarez} the model parameters were fitted to the large earthquakes in the North Anatolian Fault Zone through maximum likelihood and the Markov Poisson point process assumption was tested.
In order to capture a non monotonic behavior in the hazard, in \cite{GaravagliaPavani} the model of Alvarez was modified and a Markov Renewal process with inter-occurrence times that are mixtures of an exponential and a Weibull distribution was fitted to the same Turkish data.
In \cite{Masala} a parametric semi-Markov model with a generalized Weibull distribution for the inter-occurrence times was adapted to Italian earthquakes. Actually the semi-Markov model with generalized Weibull distributed times was first used in \cite{Foucher} to study the evolution of HIV infected patients. \cite{Votsi} considered a semi-Markov model for the seismic hazard assessment in the Northern Aegean sea and estimated the quantities of interest (semi-Markov kernel, Markov Renewal functions, etc.) through a nonparametric method.
While a wide literature concerning classical inference for Markov Renewal models for earthquake forecasting exists, to our knowledge a Bayesian approach is limited in this context. \cite{PatwardhanEtAlii} considered a semi-Markov model with log-normal distributed discrete inter-occurrence times and applied it to the large earthquakes in the circum-Pacific belt.
They stressed the fact that it is relevant to use Bayesian techniques when prior knowledge is available and it is fruitful even if the sample size is small.
\cite{MarinPlaRiosInsua} also employed semi-Markov models in the Bayesian framework, applied to a completely different area: sow farm management. They used WinBugs to perform computations (but without giving details) and they elicited their prior distributions on parameters from knowledge on farming practices.
From a probabilistic viewpoint, a Bayesian statistical treatment of a semi-Markov process amounts to model the data as a mixture of semi-Markov processes, where the mixing measure is supported on the parameters, by means of their prior laws. A complete characterization of such a mixture has been given in
\cite{EpifaniFortiniLadelli}.
In this paper we develop a parametric Bayesian analysis for a Markov Renewal process modelling earthquakes in an Italian seismic region. The magnitudes are classified into three categories according to their severity: low, medium and high size, and these categories represent the states visited by the process. As in \cite{Alvarez}, the inter-occurrence times are assumed to be Weibull random variables. The ``current sample'' is formed by the sequences of earthquakes in a homogeneous seismic region and by the corresponding inter-occurrence times collected up to a time $T$. When $T$ does not coincide with an earthquake, the last observed inter-occurrence time is censored.
The prior distribution of the parameters of the model is elicited using a ``learning dataset'', i.e. data coming from a seismic region similar to that under analysis.
The posterior distribution of the parameters is obtained through Gibbs sampling and the following summaries are estimated: transition probabilities, shape and scale parameters of the Weibull holding times for each transition and the so-called cross state-probabilities (CSPs). The transition probabilities indicate whether the strength of the next earthquake is in some way dependent on the strength of the last one; the shape parameters of the holding times indicate whether the hazard rate between two earthquakes of given magnitude classes is decreasing or increasing; the CSPs give the probability that the next earthquake occurs at or before a given time and is of a given magnitude, conditionally on the time elapsed since the last earthquake and on its magnitude.
The paper is organized as follows. In Section~\ref{sec:1} we illustrate the dataset and we discuss the choice of the Weibull model in detail. Section~\ref{sec:2} introduces the parametric Markov Renewal model.
Section~\ref{sec:3} deals with the elicitation of the prior.
Section~\ref{sec:apennines} contains the Bayesian data analysis with the estimation of the above-mentioned summaries. We also test a time predictable and a slip predictable model against the data. Section~\ref{sec:6} is devoted to some concluding remarks. Appendix \ref{sec:gibbsampling} contains the detailed derivation of the full conditional distributions and the JAGS (Just Another Gibbs Sampler) implementation of the Gibbs sampler (\cite{Plummer}).
\section{A test dataset}
\label{sec:1}
We tested our method on a sequence of seismic events chosen among those examined in \cite{Rotondi}, which was given us by the author. The sequence collects events that occurred in a tectonically homogeneous macroregion, identified as $\text{MR}_3$ by Rotondi and corresponding to the central Northern Apennines in Italy. The subdivision of Italy into eight (tectonically homogeneous) seismic macroregions can be found in the \cite{DISS} and the data are collected in the \cite{CPTI04} catalogue. If one considers earthquakes with magnitude\footnote{We refer to the moment magnitude which is related to the seismic moment $M_0$ by the following relationship: $M_w = \frac{2}{3}(\log_{10}M_0-16.05$); see \cite{Hanks}, where it is denoted by \textbf{M}.} $M_w\ge 4.5$, the sequence is complete from year 1838: a lower magnitude would make the completeness of the series questionable, especially in its earlier part. The map of these earthquakes marked by dots appears in Figure~\ref{fig:mappa}.
\begin{figure}
\caption{Map of Italy with dots indicating earthquakes with magnitude $M_w\ge 4.5$ belonging to macroregion $\text{MR}
\label{fig:mappa}
\end{figure}
As a lower threshold for the class of strong earhquakes, we choose $M_w\ge 5.3$, as suggested by \cite{Rotondi}. Then a magnitude state space with three states is obtained by indexing an earthquake by 1, 2 or 3 if its magnitude belongs to intervals $[4.5,4.9)$, $[4.9,5.3)$, $[5.3,+\infty)$, respectively. Magnitude 4.9 is just the midpoint between 4.5 and 5.3 and the released energy increases geometrically as one moves through the endpoints, with a common ratio of 4: if $M_0(M_w)$ denotes the seismic moment $M_0$ associated with $M_w$, then $M_0(5.3)/M_0(4.9) = M_0(4.9)/M_0(4.5) = 10^{\frac{3}{2} 0.4} \simeq 4$.
The energy released from an earthquake with $M_w=4.9$ does not match the midpoint between seismic moments associated with magnitudes $4.5$ and $5.3$ (in fact, this correspondence holds if $M_w=5.1$). However, there seem to be no general rule in the literature for splitting magnitude intervals. For example, \cite{Votsi} used cut-points 5.5, 5.7 and 6.1, so that $M_0(6.1)/M_0(5.7)\simeq 4$ and $M_0(5.7)/M_0(5.5)\simeq 2$, while the energy midpoint is at $M_w=5.9$; following \cite{Altinok}, \cite{Alvarez} uses cut-points 5.5, 6.0 and 6.5; \cite{Masala} employed the magnitude classes $M_w<4.7$, $M_w\in [4.7, 5)$, $M_w\ge 5$. All these authors do not give any special reason for their choices.
A more structured approach is attempted by \cite{Sadeghian}, who applied a statistical clustering algorithm to magnitudes, and again by \cite{Votsi} when they propose a different classification of states that combines both magnitude and fault orientation information. From a modelling viewpoint, this latter approach is certainly preferable, because it is likely to produce more homogeneous classes, however we do not have enough additional information to attempt this type of classification of our data in a meaningful way. An entirely different approach is that based on risk, in which cut-points would change with the built environment.
We now examine inter-occurrence times. \cite{Rotondi} considers a nonparametric Bayesian model for the inter-occurrence times between strong earthquakes (i.e. $M_w\ge 5.3$), after a preliminary data analysis which rules out Weibull, Gamma, log-normal distributions among others frequently used.
On the other hand, with a Markov Renewal model, the sequence of all the inter-occurrence times is subdivided into shorter ones according to the magnitudes, so that we think that a parametric distribution is a viable option. In particular,
we focussed on the macroregion $\text{MR}_3$ because the Weibull distribution seems to fit the inter-occurrence times better than in other macroregions. This fact is based on qq-plots. The qq-plots for $\text{MR}_3$ are shown in Figure~\ref{fig:qqplots}. The plot for transitions from 1 to 3 shows a sample quantile that is considerably larger than expected. The outlying point corresponds to a long inter-occurrence time of about 9 years, between 1987 and 1996, while 99~percent of the inter-occurence times are below 5 years. Obviously, the classification into macroregions influences the way the earthquake sequence is subdivided.
Given the Markov Renewal model framework, holding time distributions other than the Weibull could be used, such as the inverse Gaussian, the log-normal and the Gamma. However, the inverse Gaussian qq-plots clearly indicate that this distribution does not fit the data. As for the log-normal, the outlying point in the qq-plot of the $(1,3)$ transition becomes only a little less isolated, but at the expense of introducing an evident curvature in the qq-plot of the $(1,1)$ transition, whereas the remaining qq-plots are unchanged. The Gamma qq-plots are indistinguishable from the Weibull qq-plots, but we prefer working with the Weibull in view of the existing literature on seismic data analysis where the Weibull is employed. In this respect, we could follow \cite{Masala} and choose the generalized Weibull, which includes the Weibull, but the qq-plots are unchanged even with the extra parameter.
From a Bayesian computational point of view, there is no special reason for preferring the (possibly generalized) Weibull to the Gamma, as neither of them possesses a conjugate prior distribution and numerical methods are needed in both cases for making inference.
In the existing literature, the Weibull distribution has been widely used to model holding times between earthquakes from different areas and with different motivations. In Section \ref{sec:introduction} we mentioned \cite{Alvarez}, \cite{GaravagliaPavani} and \cite{Masala}, but there are also other authors. \cite{AbaimovEtAlii} argued that the increase in stress caused by the motion of tectonic plates at plate boundary faults is adequately described by an increasing hazard function, such as the Weibull can have. Instead, other distributions have an inappropriate tail behaviour: the log-normal hazard tends to zero with time and the inverse Gaussian hazard tends to a constant. Goodness-of-fit checks for the recurrence times of slip events in the creeping section of the San Andreas fault in central California confirmed that the Weibull is preferable to the mentioned alternatives. \cite{HrostMousl} considered a Weibull model, for single faults (or fault systems with homogeneous strength statistics) and power law stress accumulation. They derived the Weibull model from a theoretical framework based on the statistical mechanics of brittle fracture and they applied it to microearthquake sequences (small magnitudes) from the island of Crete and from a seismic area of Southern California, finding agreement with the data except for some deviations in the upper tail. Regarding tail behaviour, we can make a connection with
\cite{HasumiEtAlii}, who analyzed a catalogue of the Japan Meteorological Agency. These data support the hypothesis that the holding times can be described by a mixture of a Weibull distribution and a log-Weibull distribution (which possesses a heavier tail); if only earthquakes with a magnitude exceeding a threshold are considered, the weight of the log-Weibull component becomes negligible as the threshold increases.
\begin{figure}
\caption{Weibull qq-plots of earthquake inter-occurrence times (central Northern Appennines) classified by transition between magnitude classes.}
\label{fig:qqplots}
\end{figure} | 3,395 | 34,990 | en |
train | 0.32.2 | \section{Markov Renewal model}
\label{sec:2}
Let us observe, over a period of time $[0,T]$, a process in which different events occur, with random inter-occurence times.
Let us suppose that the possible states of the process are the points of a finite set
$E=\{1,\dots , s \}$ and that the process starts from state $j_0$.
Let us denote by $\tau$ the number of times the process changes states in the time interval $[0,T]$ and by
$t_i$ the time of the $i$-th change of state. Hence,
$0< t_1< \dots < t_{\tau}\leq T$. Let $j_0, j_1,\dots, j_{\tau}$ be the sequence of states visited by
the process and $x_i$ the holding time in the state $j_{i-1}$, for $i=1,\dots, \tau$. Then
\begin{equation*}
x_i= t_i-t_{i-1} \qquad \mbox{for} \; i=1,\dots, \tau
\end{equation*}
with $t_0:=0$. Furthermore, let $u_T$ be the time spent in $j_\tau$
\begin{equation*}
u_T= T-t_{\tau} ,
\end{equation*}
so the time $u_T$ is a right-censored time.
Finally, our data are collected in the vector $(\bj, \bx, u_T)$, where $ (\bj, \bx) =(j_{n}, x_{n})_{n=1,\ldots, \tau }$.
In what follows, we assume that the data $(\bj, \bx, u_T)$ are the result of the observation
of a homogeneous Markov Renewal process $(J_n, X_n)_{n\geq 0}$ starting from $j_0$.
This means that the sequence $(J_n, X_n)_{n\geq 0}$ satisfies
\begin{equation}
P(J_0=j_0) = 1, \quad P(X_0=0) =1
\label{eq:MRP_J_0}
\end{equation}
and for every $n\geq 0$, $j\in E$ and $t\geq 0$
\begin{equation}
\label{eq:MRP_n}
P(J_{n+1}=j, X_{n+1}\leq t | (J_{k}, X_{k})_{k\leq n})=
P(J_{n+1}=j, X_{n+1}\leq t | (J_{n}, X_{n}))= p_{J_{n} j} F_{J_{n} j}(t) \ .
\end{equation}
The transitions probabilities $p_{ij}$'s are collected in a transition matrix $\bp = (p_{ij})_{i,j \in E}$ and $(F_{ij})_{i,j \in E}$ is an array of
distribution functions on $\mathbb{R}_{+}=(0,+\infty)$.
For more details on Markov Renewal processes see, for example, \cite{LimniosOprisan}.
We just recall that, under Assumptions~\eqref{eq:MRP_J_0} and \eqref{eq:MRP_n}:
\begin{itemize}
\item[--] the process $(J_n)_{n\geq 0}$ is a Markov chain, starting from $j_0$, with transition matrix $\bp$,
\item[--] the holding times $(X_n)_{n\geq 0}$, conditionally on $(J_n)_{n\geq 0}$, form a sequence of independent positive random variables, with distribution function $F_{J_{n-1}\ J_n}$.
\end{itemize}
We assume that the functions $F_{ij}$ are absolutely continuous with respect to the Lebesgue measure
with density $f_{ij}$. Hence, the likelihood function of the data $(\bj, \bx, u_T)$ is
\begin{equation}
\label{eq:likelihood_general}
L(\bj, \bx, u_T)=
\left(\prod_{i=0}^{\t-1}p_{j_i j_{i+1}}f_{j_i j_{i+1}}(x_{i+1})\right)^{\mathds{1}(\tau>0)}\times
\sum_{k\in E}p_{j_{\t}k} \bar{F}_{j_{\t}k}(u_T),
\end{equation}
where, for every $x$, $\bar{F}_{ij}$ is the survival function
\begin{displaymath}
\bar{F}_{ij}(x)=1-F_{ij}(x)=P(X_{n+1}>x|J_n=i, J_{n+1}=j) \ .
\end{displaymath}
Furthermore, we assume that each inter-occurrence time has a Weibull density $f_{ij}$ with
shape parameter $\a_{ij}$ and scale parameter $\theta_{ij}$, i.e.
\begin{equation}
\label{eq:weib}
f_{ij}(x)=\frac{\a_{ij}}{\theta_{ij}}\left(\frac{x}{\theta_{ij}}\right)^{\a_{ij}-1}
\exp\left\{-\left(\frac{x}{\theta_{ij}}\right)^{\a_{ij}}\right \} \ , \quad x>0 , \; \a_{ij}>0, \;\theta_{ij}>0 \ .
\end{equation}
For conciseness, let
${\boldsymbol{\alpha}}=(\a_{ij})_{i,j\in E}$ and ${\boldsymbol{\theta}}=(\th_{ij})_{i,j\in E}$.
In order to write the likelihood in a more convenient way, let us introduce the following natural statistics. We will say that the process visits the string $(i,j)$ if a visit to $i$ is followed by a visit to $j$ and we denote by
\begin{itemize}
\item[--] $x^{\r}_{ij}$ the time spent in state $i$ at the $\r$-th visit to the string $(i,j)$,
\item[--] $N_{ij}$ the number of visits to the string $(i,j)$.
\end{itemize}
Then, assuming $\tau\ge 1$, Equations~\eqref{eq:likelihood_general} and \eqref{eq:weib} yield the following
representation of the likelihood function
\begin{multline}
\label{eq:likelihood-param}
L(\bj, \bx, u_T \, |\, \bp, \boldsymbol{\a},\boldsymbol{ \theta}) =\prod_{i,k\in E}p_{ik}^{N_{ik}}\times \\
\times \prod_{i,k\in E}\left[\a_{ik}^{N_{ik}}\frac{1}{\theta_{ik}^{\a_{ik}N_{ik}}}\left(\prod_{\rho=1}^{N_{ik}}x^{\r}_{ik}\right)^{\a_{ik}-1} \times
\exp\left\{-\frac{1}{\theta_{ik}^{\a_{ik}}}\sum_{\rho =1}^{N_{ik}}(x^{\r}_{ik})^{\a_{ik}}\right\}\right]\times \\
\times \left(\sum_{k\in E}p_{j_{\t}k}
\exp\left\{-\left(\frac{u_T}{\theta_{j_{\t} k}}\right)^{\a_{j_{\t} k}}\right\}\right)\ .
\end{multline}
Our purpose is now to perform a Bayesian analysis for $\bp, \boldsymbol{\alpha}$ and $ \boldsymbol{\theta}$ which allows us to introduce prior knowledge on the parameters. As shown in Appendix \ref{sec:gibbsampling}, this analysis is possible via a Gibbs sampling approach. | 1,885 | 34,990 | en |
train | 0.32.3 | \section{Bayesian analysis}
\label{sec:3}
\subsection{The prior distribution}
\label{subsec:3.1}
Let us assume that a priori $\bp$ is independent of $\ba$ and $\bth$. In particular,
the rows of $\bp$ are $s$ independent vectors with Dirichlet distribution
with parameters $\bg_1, \cdots ,\bg_s$ and total mass $c_1, \cdots,c_s$, respectively. This means
that, for $i=1,\ldots ,s$, the prior density of the $i$-th row is
\begin{equation}
\label{eq:dirich}
\pi_{1,i}(p_{i1},\ldots , p_{is}) = \frac{\Gamma(c_i)}{\prod_{j=1}^{s}\Gamma(\g_{ij})} {\prod_{j=1}^{s}p_{ij}^{\g_{ij}-1}}
\end{equation}
on $T=\{ (p_{i1},\ldots , p_{is})| \ p_{ij} \geq 0,\ \sum_j p_{ij} =1\}$ where $\bg_i =(\g_{i1}, \cdots , \g_{is})$, with $\g_{ij}>0$ and $c_i=\sum_{j=1}^s \g_{ij}$.
As far as $ \ba$ and $\bth$ are concerned, the $\th_{ij}$'s, given the $\a_{ij}$'s, are independent with generalized inverse Gamma
densities
\begin{equation}
\label{eq:pi2}
\pi_{2,ij}(\th_{ij}| \ba) = \pi_{2,ij}(\th_{ij}| \a_{ij}) =
\frac{\a_{ij}b_{ij}(\a_{ij})^{m_{ij}}}{\Gamma(m_{ij})}\th_{ij}^{-(1+m_{ij}\a_{ij})}
\times \exp\left\{-\frac{b_{ij}(\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\},
\;\; \th_{ij}>0,
\end{equation}
where $m_{ij}>0$ and
\begin{equation}
\label{eq:b-q}
b_{ij}(\a_{ij}) = \left(t^{ij}_{q_{ij}}\right)^{\a_{ij}}[(1-q_{ij})^{-1/m_{ij}}-1]^{-1}
\end{equation}
with
$t^{ij}_{q_{ij}}>0$ and $q_{ij}\in (0,1)$.
In other terms, $\th_{ij}^{- \a_{ij}}$, given $ \a_{ij}$, has a prior Gamma density with shape $m_{ij}$ and scale $1/b_{ij}(\a_{ij})$. In symbols $\th_{ij}|
\a_{ij} \sim m_{\alpha}thcal{ GIG}(m_{ij}, b_{ij}(\a_{ij}), \a_{ij})$.
We borrow the expression of the $b_{ij}(\a_{ij})$'s in \eqref{eq:b-q} from \cite{Bousquet2} and, as a consequence of this choice, $t^{ij}_{q_{ij}}$ turns out to be the marginal quantile of order $q_{ij}$ of an inter-occurrence time between states $i$ and $j$. Indeed, if $\pi_{3,ij}$ denotes the density of $\a_{ij}$ and $X$ is such a random time, then
\begin{align*}
P(X> t) & = \int_{0}^{+\infty}\int_{0}^{+\infty}
P(X> t|\a_{ij},\theta_{ij})\pi_{2,ij}(\th_{ij}|\a_{ij})\pi_{3,ij}(\a_{ij})d\th_{ij} d\a_{ij}\\
& = \int_{0}^{+\infty}\Big[\frac{b_{ij}(\a_{ij})}{b_{ij}(\a_{ij})+t^{\a_{ij}}}\Big]^{m_{ij}}\pi_{3,ij}(\a_{ij})d\a_{ij} \ ,
\qquad \qquad \qquad \qquad\;\forall t>0.
\end{align*}
Hence, in view of \eqref{eq:b-q}, if $t=t^{ij}_{q_{ij}}$, we obtain $P(X>t^{ij}_{q_{ij}})=1-q_{ij}$, for every proper prior density $\pi_{3,ij}$.
Finally, a priori, the
components of $ \ba$ are independent and have
densities $\pi_{3,ij}$
such that
\begin{multline}
\label{eq:pi3}
\pi_{3,ij}(\a_{ij})\propto
\a_{ij}^{m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}\exp\{-m_{ij}d_{ij}\a_{ij}\}\mathds{1}(\a_{ij}\ge\a_{0,ij}), \\
\a_{0,ij}\geq 0,\; c_{ij}> 0,\; m_{ij}>0, \; d_{ij}\ge 0\ .
\end{multline}
As far as the prior $\pi_{3,ij}$ is concerned, it is easy to see that:
\begin{enumerate}[$a)$]
\item if $d_{ij} >0$, then $\pi_{3,ij}$ is a proper prior;
\item if $\a_{0,ij}=0$ and $d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density;
\item if $c_{ij}=1 , \ \a_{0,ij}>0 \mbox{ and } d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density
truncated from below at $\a_{0,ij}$;
\item if $c_{ij}=m_{ij}, \ \a_{0,ij}>0 \mbox{ and } d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density shifted by
$\a_{0,ij}$;
\item if $c_{ij}=1$ and $m_{ij}\to 0$, then $\pi_{2,ij}(\theta_{ij}|\a_{ij})\pi_{3,ij}(\a_{ij})$ approaches the Jeffreys prior for the Weibull model: $1/\theta_{ij}\mathds{1}{(\theta_{ij}>0)}\mathds{1}{(\alpha_{ij} \geq \a_{0,ij})}$;
\item if $c_{ij} \ge 1$ \mbox{ and } $m_{ij}\ge 1$, then $\pi_{3,ij}$ is a log-concave function.
\end{enumerate}
The prior corresponding to the choices in $c)$ was first introduced in \cite{Bousquet} and \cite{Bousquet2}.
As discussed in \cite{GilksWild}, the log-concavity of $\pi_{3,ij}$ is necessary
in the implementation of the Gibbs sampler (see also \cite{BergerSun}), although adjustments exist for the non-log-concave case (see \cite{GilksBestTan}).
Furthermore, we will show later that a support suitably bounded away from zero ensures the existence of the posterior moments of the $\th_{ij}$'s.
\subsection{Elicitation of the hyperparameters}
In this section we focus our attention on the prior of $ ( \a_{ij},\th_{ij})$, for fixed $i,j$. Adapting the approach developed by Bousquet to our situation, we give a statistical justification of the prior introduced in Subsection~\ref{subsec:3.1}. An interpretation of the hyperparameters is also provided.
For the sake of semplicity, let us drop the indices $i,j$ in all the notations and quantities.
Suppose that a ``learning dataset'' $\by_{m} = (y_{1},\ldots , y_{m})$ of $m$ holding times in the state $i$ followed by a visit to the state $j$ is available from another seismic region similar to the one under analysis.
Therefore the prior scheme defined by Equations \eqref{eq:pi2}--\eqref{eq:pi3} can be interpreted as a suitable modification of a posterior distribution of $( \a, \th)$, given the learning dataset $\by_{m}$.
This approach allows us to elicit the hyperparameters.
More precisely, consider
for $(\a, \theta)$ the posterior density, conditionally on $\by_{m}$, when we start from the following improper prior:
\begin{equation}
\label{eq:pi_alpha1}
\tilde{\pi}(\a, \theta) \propto \theta^{-1}\left(1-\frac{\alpha_0}{\alpha}\right)^{c-1} \mathds{1}{(\theta \geq 0)} \mathds{1}{(\alpha \geq \alpha_0)} \ ,
\end{equation}
for some suitable $c\geq 1$ and $\alpha_0\geq 0$ (The condition $c\geq 1$ guarantees that $\tilde{\pi}(\a, \theta)$ is a log-concave function with respect to $\alpha$).
Consequently, the posterior density of $\theta$, given $\alpha$ and $\by_{m}$, is
\begin{equation}
\label{eq:prior-theta}
\tilde\pi_2(\theta| \by_{m}, \alpha ) = m_{\alpha}thcal{ GIG}(m, \tilde b(\by_{m} ,\alpha), \alpha)
\end{equation}
and the posterior density of $\alpha$ is
\begin{equation}
\label{eq:prior-alpha}
\tilde\pi_3(\alpha |\by_{m}) \propto \frac{\alpha^{m-c}(\alpha -\alpha_0)^{c-1}}{\tilde b^m(\by_{m}, \alpha)}
\exp\{{-m}{\beta(\by_{m})\alpha}\}\mathds{1}{(\alpha \geq \alpha_0)} \ ,
\end{equation}
with $\tilde b( \by_{m},\alpha)=\sum_{i=1}^{m}y^{\alpha}_{i}$ and $\beta(\by_{m})= {\sum_{i=1}^{m}\ln y_{i}}/{m}$.
Notice that the posterior we obtain
has a simple hierarchical structure: $\tilde\pi_2(\theta|\by_{m},\alpha)$ is a generalized inverse Gamma density and this provides both a justification of the form of the $\pi_2(\theta|\alpha)$ in \eqref{eq:pi2} and an interpretation of the first parameter $m$. Indeed $m$ is equal to the size of the learning dataset $\by_m$ and so it is a measure of prior uncertainty.
Now, if we replace the function $\tilde b(\by_{m},\alpha )$ in \eqref{eq:prior-theta} and \eqref{eq:prior-alpha} by the easier convex function of $\alpha$ introduced in \eqref{eq:b-q}, i.e.
$b(\alpha)= t^{\alpha}_q[(1-q)^{-1/m}-1]^{-1}$, with $t_q>0$ and $q\in (0,1)$,
then $\tilde\pi_3(\alpha |\by_{m})$ takes the same form as
in \eqref{eq:pi3} with
\begin{equation}
\label{eq:d_ij}
d=\ln {t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m} \ .
\end{equation}
In this way, we obtain a justification of the form of the prior densities $\pi_{3,ij}$'s in \eqref{eq:pi3} and an easy way to elicit its parameter $d_{ij}$ when the learning dataset is available.
Furthermore, $b(\alpha)$ can be also elicited once the predictive quantile $t_q$ is specified. Its specification can be accomplished, for example, in the two following different ways:
\begin{enumerate}
\item we estimate an empirical quantile $\hat t_q$ from the learning dataset;
\item an expert is asked about the chance, quantified by $q$, of an earthquake before $t_q$.
\end{enumerate}
\mbox{}\indent In the following, if a learning dataset of size $m\ge 2$ is available, we consider an empirical quantile $\hat{t}_q$ of order $q$ such that
\begin{displaymath}
\ln \hat{t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m}> 0\ .
\end{displaymath}
Therefore, letting $\hat{b}(\alpha)$ denote the value of $b(\a)$ corresponding to $\hat{t}_q$, we propose a Bayesian analysis based on the prior
\begin{equation}\label{pi-2}
\pi_2(\theta|\alpha )=m_{\alpha}thcal{ GIG}(m, \hat{b}(\alpha), \a)
\end{equation}
and
\begin{equation}\label{pi-3}
\pi_3(\alpha)\propto \a^{m-c}(\alpha-\a_0)^{c-1} \exp\left\{-m\left(\ln \hat{t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m}\right)\a\right\}\mathds{1}{(\alpha\geq \a_0)} \ ,
\end{equation}
where $m$ is the size of the learning dataset. In addition, we choose $c=m$ so that $\pi_3(\alpha)$ is a shifted Gamma prior and consequently it is proper and log-concave.
The remaining hyperparameter $\a_0$ is chosen so that the posterior second moment of $\theta$ is finite. If $\a$ is bounded away from zero, then
\begin{displaymath}
\E(\theta^2) = \E\left(\frac{\Gamma(m-2/\a)}{\Gamma(m)}\left[\hat{b}(\alpha)\right]^{2 / \a}\right)\leq \tilde{K} \E(\Gamma(m-2/\a))
\end{displaymath}
for a suitable constant $\tilde{K}$. As a consequence if $\a_0=2/m$, then $\E(\theta^2)<+\infty$ and hence also the posterior second moment of $\theta$ is finite.
The choice $\a_0=2/m$ is suitable only if $m>2$.
If $m=2$, then
$\a_0=2/m=1$ and decreasing hazard rates are ruled out.
In the absence of additional specific prior information, this is an arbitrary restriction, so a value for $\a_0$ smaller than 1 must be chosen. Then, the prior second moment of $\theta$ is not finite anymore. On the other hand, for the posterior second moment to be finite, we need $\alpha>2/(2+N)$, where $N$ is the number of transitions between the two concerned states in the (current) sample. Thus the second moment of $\theta$ can stay non-finite, even a posteriori, if $2/(2+N)>\alpha_0$. This would show that the data add little information for that specific transition. To avoid this, we may let $\a_0$ be the minimum between the value $2/3$, corresponding to the smallest learning sample size such that $\a_0<1$, and the value $2/(2+N)$, necessary for the finiteness of the posterior second moment. Therefore, $\displaystyle \alpha_0 = \min\{2/3, 2/(2+N)\}$.
Finally, if $\gamma$ denotes the hyperparameter corresponding to the indexes $i$ and $j$ in the Dirichlet prior \eqref{eq:dirich}, then we select $\gamma=m+1$, i.e. $\gamma$ is equal to the number of transitions from state $i$ to state $j$ in the learning dataset, plus one. | 3,875 | 34,990 | en |
train | 0.32.4 | \subsection{Scarce prior information}\label{sec:scarce}
The construction of the prior distribution of $(\alpha,\theta)$ must be modified for those pairs of states between which no more than one transition was observed in the learning dataset.
If $m=1$, the single learning observation $y_{1}$ determines $\hat{b}(\alpha)$. As $\hat{t}_q=y_{1}$ for any $q$, it seems reasonable to use $q=0.5$, so $y_{1}$ would represent the prior opinion on the median holding time. Since $d=0$ when $m=1$, then $\pi_3(\alpha)$ is improper for any $c>0$. We make it proper by restricting its support to an interval $(\alpha_0,\alpha_1)$. The value $\alpha_1=10$ is suitable for all practical purposes. As before, the choice $\alpha_0 = 2/m=2$ would be too much restrictive, so we select again $\displaystyle \alpha_0 = \min\{2/3, 2/(2+N)\}$. With regard to $c$, we put $c=2$.
Furthermore, the elicitation of the hyperparameter of the Dirichlet prior is again $\gamma = m+1=2$, i.e. the number of transitions observed in the learning dataset (just one) plus one.
If $m=0$, the prior information on the number of transitions is that there have been no transitions, but there is no information on the holding times.
In this case we can represent in the model the absence of information, choosing
\[
q=0.5\quad\mbox{and}\quad \tilde t_{0.5}\sim U(t_1,t_2),
\]
that is $\tilde t_{0.5}$ is uniformly distributed over a big time interval $(t_1,t_2)$, independently from everything else.
Hence, we use $q=0.5$ and $\tilde{t}_{0.5}$ to obtain $\hat{b}(\alpha)$ and we fall in the previous case by substituting $m=1$ to $m=0$.
For clearness, in Table~\ref{table:hyperparameters}, we summarize the hyperparameter selection for priors \eqref{eq:dirich}, \eqref{pi-2} and \eqref{pi-3}.
\begin{table}[h]
\renewcommand*{\arraystretch}{1.4}
\begin{tabular}{r|cccc}
& $m>2$ & $m=2$ & $m=1$ & $m=0$\\
\hline
$t_q$ & $\hat{t}_q$ & $\hat{t}_q$ & $y_{1}$ & $\tilde{t}_q\sim U(t_1,t_2)$\\
$c$ & $m$ & $m$ & 2 & 2\\
$\a_0$ & $\frac{2}{m}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ \\
$\gamma$ & $m+1$ & $m+1$ & $m+1$ & $2$ \\
\end{tabular}
\caption{Hyperparameter selection as the learning sample size $m$ varies.}
\label{table:hyperparameters}
\end{table} | 815 | 34,990 | en |
train | 0.32.5 | \section{Analysis of the central Northern Apennines sequence}\label{sec:apennines}
In this section we analyze the macroregion $\text{MR}_3$ sequence, using the semi-Markov model.
We coded the Gibbs sampling algorithm in the JAGS software package, which is designed to work
closely with the \cite{Rpackage} package, in which all statistical computations and
graphics were performed. Details of the Gibbs sampler are in Appendix \ref{sec:gibbsampling}. On the whole, 750,000 iterations for one chain were run for estimating the unknown
parameters in the model, and the first 250,000 were discarded as burn-in.
After the burn-in, one out of every 100 simulated values was kept for
posterior analysis, for a total sample size of 5,000. The
convergence diagnostics, such as those available in the R package CODA
(Geweke, Heidelberger and Welch stationarity test, interval halfwidth
test), were computed for all parameters, indicating that convergence
has been achieved.
Model fitting, model validation and an attempt at forecasting involve the following steps:
\begin{enumerate}
\item the learning dataset for the elicitation of the prior distribution is chosen;
\item model fit is assessed by comparing observed inter-occurrence times (grouped by transition) to posterior predictive intervals;
\item cross state-probabilities are estimated, as an indication to the most likely magnitude and time to the next event, given information up to the present time;
\item an interpretation in terms of slip predictable or time predictable model is provided.
\end{enumerate}
For the elicitation of the prior distribution, the learning data are taken from $\text{MR}_4$, another macroregion among those considered by \cite{Rotondi}, who examines statistical summaries of the holding times and suggests that $\text{MR}_4$ could be used as a learning set for the hyperparameters of $\text{MR}_3$. \cite{PeruggiaSantner}, in their analysis of the magnitudes and of the inter-occurence times of eartquakes from another Italian area, chose a subset of the incomplete older part of their series to elicit prior distributions. This procedure is justifiable in their case because the old and the new part of the series can be regarded as two different processes and the cut-point between them appears to be clearly identified. If we did the same with our series, we would alter the Bayesian learning process, because we would obtain different posterior distributions on changing the cut-point position.
Transition frequencies and median inter-occurrence times appear in Table \ref{table:empirical.P} for both the $\text{MR}_3$ and the $\text{MR}_4$ datasets. The Dirichlet hyperparameters $\bg_1, \cdots ,\bg_s$ are equalled to the rows of Table \ref{table:empirical.P}\subref{table:0.empirical.P} plus one. The medians are reported because we have selected $q=0.5$ in Table \ref{table:hyperparameters}: the medians in Table \ref{table:empirical.P}\subref{table:0.median} are smaller than the medians in Table \ref{table:empirical.P}\subref{table:1.median} in six entries out of nine, in some cases considerably.
\begin{table}[h]
\begin{center}
\subtable[\label{table:1.empirical.P}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 65 & 30 & 17 \\
from 2 & 32 & 15 & 7 \\
from 3 & 15 & 9 & 4 \\
\hline
\end{tabular}
}}\quad
\subtable[\label{table:0.empirical.P}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 114 & 51 & 13 \\
from 2 & 56 & 25 & 4 \\
from 3 & 8 & 8 & 3 \\
\hline
\end{tabular}
}}\quad\\
\subtable[\label{table:1.median}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 204 & 257 & 141 \\
from 2 & 150 & 122 & 219 \\
from 3 & 142 & 82 & 309 \\
\hline
\end{tabular}
}}\quad
\subtable[\label{table:0.median}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 105 & 61 & 193 \\
from 2 & 104 & 99 & 76 \\
from 3 & 209 & 117 & 78 \\
\hline
\end{tabular}
}}
\end{center}
\caption{Summaries of datasets $\text{MR}_3$ (tables on the left) and $\text{MR}_4$ (tables on the right); $\text{MR}_4$ is the learning dataset used for hyperparameter elicitation. (a) and (b): number of observed transitions; (c) and (d): median inter-occurrence times (in days).}
\label{table:empirical.P}
\end{table}
Let us consider the predictive check mentioned above. Figure~\ref{figure:caterpillar_times} shows posterior predictive 95~percent probability intervals of the inter-occurrence times for every transition, with the observed inter-occurrence times superimposed. These are empirical intervals computed by generating stochastic inter-occurrence times from their relevant distributions at every iteration of the Gibbs sampler. Possible outliers, represented as triangles, are those times with Bayesian $p$-value (that is the predictive tail probability) less than 2.5~percent.
\begin{figure}
\caption{Posterior predictive $95$~percent credible intervals of the inter-occurrence times in days with actual times denoted by (blue) solid dots. Suspect outliers are denoted by (red)-pointing triangles. The (green) dotted line shows the posterior median and the (violet) dashed line the posterior mean. The prior distribution was elicited from the $\text{MR}
\label{figure:caterpillar_times}
\end{figure}
In Table~\ref{table:posterior.time} we report the expected value (and the standard deviation) of the inter-occurrence times. In Table \ref{table:b.outliers} the numbers of upper and lower extreme points and their overall percentage are collected. While deviations from the nominal 95\% coverage are acceptable for transitions with low absolute frequency, such as $(2,3)$, $(3,3)$, $(3,2)$, the remaining transitions require attention. We see that the percentage of outliers higher than the nominal value is mostly due to the upper outliers, which occur as an effect of the difference between the prior opinion on the marginal median of the inter-occurrence times and the median of the observed sequence (compare Table \ref{table:empirical.P}\subref{table:0.median} to Table \ref{table:empirical.P}\subref{table:1.median}). A
few really extreme inter-occurrence times, such as the small values observed at transitions $(1,1)$, $(2,1)$ and the large one at transition $(1,3)$, match unsurprisingly the outlying points in the corresponding qq-plots in Figure~\ref{fig:qqplots}. This fact could be regarded as a lack of fit of the Weibull model, but it could also be due to an imperfect assignment of some events to the macroregion $\text{MR}_3$ or to an insufficient filtering of secondary events (i.e. aftershocks and foreshocks): earthquakes incorrectly assigned to $\text{MR}_3$ and aftershocks or foreshocks can give rise to very short inter-occurrence times; on the other hand, earthquakes which should be in $\text{MR}_3$ but which were attributed to other macroregions can produce very long inter-occurrence times.
\begin{table}[ht]
\begin{center}
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1 &191 (12) & 172 (18) & 331 (70)\\
2 &214 (22) & 238 (43) & 354 (145)\\
3 & 263 (58) & 203 (55)& 314 (134)\\
\hline
\end{tabular}
\end{center}
\caption{Predictive means (and standard deviations) of inter-occurrence times for each transition (in days); prior elicited from $\text{MR}_4$.}
\label{table:posterior.time}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{cccc|ccc|ccc}
& \multicolumn{3}{c}{Upper outliers} & \multicolumn{3}{c}{Lower outliers} & \multicolumn{3}{c}{\% of outliers}\\
\hline
& 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3\\
\hline
1 & 8 & 3 & 1 & 2 & 0 & 0 & 15.4\% & 10.0\% & 5.9\%\\
2 & 4& 3& 1& 1& 0 & 0 & 15.6\% &20.0\% & 14.3\%\\
3 & 1 & 0 & 0 & 0 & 1 & 0 & 6.7\% & 11.1\%& 0.0\%\\
\hline
\end{tabular}
\end{center}
\caption{Number of points having lower or upper posterior $p$-value less than $2.5$~percent and their percentage; prior elicited from $\text{MR}_4$.}
\label{table:b.outliers}
\end{table}
The shape parameters $\alpha_{ij}$ are particularly important as they reflect an increasing hazard if larger than 1, a decreasing hazard if smaller than 1 and a constant hazard if equal to 1. Table~\ref{table:posterior.alpha-theta}~\subref{table:posterior.alpha} displays the posterior means of these parameters (along with their posterior standard deviations). Finally Table~\ref{table:posterior.P} shows the posterior means of the transition probabilities. Notice that the last row departs from the other two; we will return to this in the following.
\begin{table}[ht]
\centering
\subtable[Shape parameter $\alpha$ \label{table:posterior.alpha}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1 & 1.18 (0.06) & 1.07 (0.08) & 0.94 (0.10) \\
2 & 1.07 (0.07) & 0.95 (0.10) & 0.89 (0.14) \\
3 & 1.04 (0.16) & 1.03 (0.16) & 1.11 (0.21)\\
\hline
\end{tabular}
}}\quad
\subtable[Scale parameter $\theta$ \label{table:posterior.theta}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1& 201.7 (13.2)& 175.6 (19.1)& 317.8 (67.0)\\
2 & 219.2 (22.5)& 231.0 (40.5)& 327.3 (132.4)\\
3 & 262.7 (57.2) & 201.9 (52.2) & 320.1 (133.5)\\
\hline
\end{tabular}
}}
\caption{Posterior means (with standard deviations) of the shape parameter $\alpha$ in \subref{table:posterior.alpha} and of the scale parameter $\theta$ in \subref{table:posterior.theta}; prior elicited from $\text{MR}_4$.}
\label{table:posterior.alpha-theta}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrr}
\hline
& 1 & 2 & 3 \\
\hline
1 & 0.614 (0.028) & 0.280 (0.026) & 0.106 (0.018)\\
2 & 0.626 (0.041) & 0.290 (0.038) & 0.085 (0.023)\\
3 & 0.479 (0.070) & 0.361 (0.067) & 0.160 (0.051)\\
\hline
\end{tabular}
\end{center}
\caption{Summaries of the posterior distributions of the transition matrix $\bp$. Posterior means (with standard deviations); prior elicited from $\text{MR}_4$.}
\label{table:posterior.P}
\end{table} | 3,513 | 34,990 | en |
train | 0.32.6 | \begin{table}[ht]
\begin{center}
\begin{tabular}{cccc|ccc|ccc}
& \multicolumn{3}{c}{Upper outliers} & \multicolumn{3}{c}{Lower outliers} & \multicolumn{3}{c}{\% of outliers}\\
\hline
& 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3\\
\hline
1 & 8 & 3 & 1 & 2 & 0 & 0 & 15.4\% & 10.0\% & 5.9\%\\
2 & 4& 3& 1& 1& 0 & 0 & 15.6\% &20.0\% & 14.3\%\\
3 & 1 & 0 & 0 & 0 & 1 & 0 & 6.7\% & 11.1\%& 0.0\%\\
\hline
\end{tabular}
\end{center}
\caption{Number of points having lower or upper posterior $p$-value less than $2.5$~percent and their percentage; prior elicited from $\text{MR}_4$.}
\label{table:b.outliers}
\end{table}
The shape parameters $\alpha_{ij}$ are particularly important as they reflect an increasing hazard if larger than 1, a decreasing hazard if smaller than 1 and a constant hazard if equal to 1. Table~\ref{table:posterior.alpha-theta}~\subref{table:posterior.alpha} displays the posterior means of these parameters (along with their posterior standard deviations). Finally Table~\ref{table:posterior.P} shows the posterior means of the transition probabilities. Notice that the last row departs from the other two; we will return to this in the following.
\begin{table}[ht]
\centering
\subtable[Shape parameter $\alpha$ \label{table:posterior.alpha}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1 & 1.18 (0.06) & 1.07 (0.08) & 0.94 (0.10) \\
2 & 1.07 (0.07) & 0.95 (0.10) & 0.89 (0.14) \\
3 & 1.04 (0.16) & 1.03 (0.16) & 1.11 (0.21)\\
\hline
\end{tabular}
}}\quad
\subtable[Scale parameter $\theta$ \label{table:posterior.theta}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1& 201.7 (13.2)& 175.6 (19.1)& 317.8 (67.0)\\
2 & 219.2 (22.5)& 231.0 (40.5)& 327.3 (132.4)\\
3 & 262.7 (57.2) & 201.9 (52.2) & 320.1 (133.5)\\
\hline
\end{tabular}
}}
\caption{Posterior means (with standard deviations) of the shape parameter $\alpha$ in \subref{table:posterior.alpha} and of the scale parameter $\theta$ in \subref{table:posterior.theta}; prior elicited from $\text{MR}_4$.}
\label{table:posterior.alpha-theta}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrr}
\hline
& 1 & 2 & 3 \\
\hline
1 & 0.614 (0.028) & 0.280 (0.026) & 0.106 (0.018)\\
2 & 0.626 (0.041) & 0.290 (0.038) & 0.085 (0.023)\\
3 & 0.479 (0.070) & 0.361 (0.067) & 0.160 (0.051)\\
\hline
\end{tabular}
\end{center}
\caption{Summaries of the posterior distributions of the transition matrix $\bp$. Posterior means (with standard deviations); prior elicited from $\text{MR}_4$.}
\label{table:posterior.P}
\end{table}
Cross state-probability plots are an attempt at predicting what type of event and when it is most likely to occur.
A cross state-probability (CSP) $P_{t_0| \Delta x}^{ij}$ represents the probability that the next event will be in state $j$ within a time interval $\Delta x$ under the assumption that the previous event was in state $i$ and $t_0$ time units have passed since its occurrence:
\begin{multline}
\label{eq:CSP}
P_{t_0| \Delta x}^{ij} = P\left(J_{n+1} =j, \ X_{n+1}\le t_0+\Delta x| \ J_n = i, \ X_{n+1} > t_0\right) = \\
= \frac{ p_{ij}\left(\bar{F}_{ij}(t_0) - \bar{F}_{ij}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{ik} \bar{F}_{ik}(t_0)} \ .
\end{multline}
Figure~\ref{figure:csp} displays the CSPs with time origin on 31 December 2002, the closing date of the \cite{CPTI04} catalogue. At this time, the last recorded event had been in class 2 and had occurred 965 days earlier (so $t_0$ is about 32 months). From these plots we can read out the probability that an event of any given type will occur before a certain number of months. For example, after 24 months, the sum of the mean CSPs in the three graphs indicates that the probability that an event will have occurred is around 88\%, with a larger probability assigned to an event of type 2, followed by type 1 and type 3. The posterior means of the CSPs are also reported in Table~\ref{table:csp}.
\begin{figure}
\caption{Posterior mean and median of CSPs with time origin on 31 December 2002 up to 48 months ahead, along with 90~percent posterior credible intervals. Transitions are from state 2 to state 1, 2 or 3 (first to third panel, respectively). Months since 31 December 2002 are along the x-axis. The learning set is $MR_4$.}
\label{figure:csp}
\end{figure}
\begin{table}[ht]
\begin{center}
\scalebox{ 0.9}{ \begin{tabular}{rrrrrrrrrrr}
\hline
&1 Month & 2 Months & 3 Months & 4 Months &5 Months & 6 Months & 1 Year& 2 Years & 3 Years & 4 Years \\
\hline
to 1 & 0.045 & 0.080 & 0.113 & 0.140 & 0.164 & 0.184& 0.256 & 0.296 & 0.303& 0.304\\
to 2 & 0.038 & 0.069 & 0.099 & 0.125 & 0.149 & 0.169& 0.257 & 0.327 & 0.348& 0.356\\
to 3 & 0.023 & 0.041 & 0.061 & 0.078 & 0.095 & 0.109& 0.180 & 0.256 & 0.291& 0.309\\
\hline
\end{tabular}}
\end{center}
\caption{CSPs with time origin on 31 December 2002, as represented in Figure~\ref{figure:csp}; prior elicited from $\text{MR}_4$.}
\label{table:csp}
\end{table}
The predictive capability of our model can be assessed by marking the time of the next event on the relevant CSP plot. In our specific case, the first event in 2003, which can be assigned to the macroregion $\text{MR}_3$ happened in the Forl\`{\i} area on 26 January and was of type 1, with a CSP of 4.5\%. This is a low probability, but a single case is not enough to judge our model, which would be a bad one if repeated comparisons did not reflect the pattern represented by the CSPs. Therefore we repeated the same comparison by re-estimating the model using only the data up to 31 December 2001, 31 December 2000, and so on backwards down to 1992. The results are shown in Table~\ref{table:31.12.92-01}. The boxed numbers correspond to the observed events and it is a good sign that they do not always correspond to very high or very low CSPs, as this would indicate that events occur too late or too early compared to the estimated model. If we were to plot the conditional densities obtained by differentiating the CSPs with respect to $\Delta x$, marking the observed inter-occurrence times on the x-axis, we would observe that very few of them appear in the tails.
\begin{table}[h!]
\centering
\scalebox{ 0.8}{\begin{tabular}{rrrrrrrrrrrr}
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2001;\quad previous event type:2;\quad holding time: 600 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 392 days & 2 Years & 3 Years & 4 Years \\
to 1 & 0.069 & 0.122 & 0.173 & 0.215 & 0.251 & 0.282 & 0.392 & \boxed{{\color{blue}0.401}} & 0.451 & 0.461 & 0.462 \\
to 2 & 0.038 & 0.068 & 0.097 & 0.123 & 0.146 & 0.166 & 0.248 & 0.256 & 0.310 & 0.328 & 0.333 \\
to 3 & 0.015 & 0.027 & 0.039 & 0.050 & 0.061 & 0.070 & 0.113 & 0.118 & 0.158 & 0.178 & 0.187 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2000;\quad previous event type: 2;\quad holding time: 235 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1Year & 2 Years & 757 days & 3 Years & 4 Years \\
to 1 & 0.085 & 0.152 & 0.217 & 0.270 & 0.318 & 0.358 & 0.505 & 0.584 & \boxed{{\color{blue}0.586}} & 0.597 & 0.599 \\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & 0.134 & 0.152 & 0.223 & 0.273 & 0.275 & 0.287 & 0.290 \\
to 3 & 0.009 & 0.017 & 0.024 & 0.031 & 0.037 & 0.042 & 0.066 & 0.090 & 0.091 & 0.099 & 0.103\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1999;\quad previous event type: 1;\quad holding time: 177 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 130 days & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.086 & 0.157 & 0.222 & 0.277 & 0.292 & 0.325 & 0.366 & 0.518 & 0.600 & 0.613 & 0.615\\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & \boxed{{\color{blue}0.119}} & 0.133 & 0.151 & 0.220 & 0.269 & 0.281 & 0.284 \\
to 3 & 0.009 & 0.016 & 0.023 & 0.029 & 0.031 & 0.035 & 0.040 & 0.062 & 0.082 & 0.091 & 0.094 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1998;\quad previous event type: 3;\quad holding time: 280 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 188 days & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.085 & 0.151 & 0.214 & 0.267 & 0.314 & 0.353 & \boxed{{\color{blue}0.361}} & 0.496 & 0.573 & 0.585 & 0.587 \\
to 2 & 0.035 & 0.063 & 0.091 & 0.114 & 0.135 & 0.153 & 0.157 & 0.225 & 0.277 & 0.290 & 0.294 \\ | 3,946 | 34,990 | en |
train | 0.32.7 | \begin{table}[h!]
\centering
\scalebox{ 0.8}{\begin{tabular}{rrrrrrrrrrrr}
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2001;\quad previous event type:2;\quad holding time: 600 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 392 days & 2 Years & 3 Years & 4 Years \\
to 1 & 0.069 & 0.122 & 0.173 & 0.215 & 0.251 & 0.282 & 0.392 & \boxed{{\color{blue}0.401}} & 0.451 & 0.461 & 0.462 \\
to 2 & 0.038 & 0.068 & 0.097 & 0.123 & 0.146 & 0.166 & 0.248 & 0.256 & 0.310 & 0.328 & 0.333 \\
to 3 & 0.015 & 0.027 & 0.039 & 0.050 & 0.061 & 0.070 & 0.113 & 0.118 & 0.158 & 0.178 & 0.187 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2000;\quad previous event type: 2;\quad holding time: 235 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1Year & 2 Years & 757 days & 3 Years & 4 Years \\
to 1 & 0.085 & 0.152 & 0.217 & 0.270 & 0.318 & 0.358 & 0.505 & 0.584 & \boxed{{\color{blue}0.586}} & 0.597 & 0.599 \\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & 0.134 & 0.152 & 0.223 & 0.273 & 0.275 & 0.287 & 0.290 \\
to 3 & 0.009 & 0.017 & 0.024 & 0.031 & 0.037 & 0.042 & 0.066 & 0.090 & 0.091 & 0.099 & 0.103\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1999;\quad previous event type: 1;\quad holding time: 177 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 130 days & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.086 & 0.157 & 0.222 & 0.277 & 0.292 & 0.325 & 0.366 & 0.518 & 0.600 & 0.613 & 0.615\\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & \boxed{{\color{blue}0.119}} & 0.133 & 0.151 & 0.220 & 0.269 & 0.281 & 0.284 \\
to 3 & 0.009 & 0.016 & 0.023 & 0.029 & 0.031 & 0.035 & 0.040 & 0.062 & 0.082 & 0.091 & 0.094 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1998;\quad previous event type: 3;\quad holding time: 280 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 188 days & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.085 & 0.151 & 0.214 & 0.267 & 0.314 & 0.353 & \boxed{{\color{blue}0.361}} & 0.496 & 0.573 & 0.585 & 0.587 \\
to 2 & 0.035 & 0.063 & 0.091 & 0.114 & 0.135 & 0.153 & 0.157 & 0.225 & 0.277 & 0.290 & 0.294 \\
to 3 & 0.010 & 0.018 & 0.025 & 0.032 & 0.039 & 0.045 & 0.046 & 0.071 & 0.096 & 0.106 & 0.111 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1997;\quad previous event type: 3;\quad holding time: 442 days}}\\
\hline
& 1 Month & 2 Months & 85 days & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1& 0.078 & 0.138 & 0.187 & 0.196 & 0.244 & 0.286 & 0.320 & 0.448 & 0.516 & 0.526 & 0.528 \\
to 2 & 0.036 & 0.065 & 0.090 & 0.094 & 0.118 & 0.140 & 0.159 & 0.237 & 0.294 & 0.310 & 0.315\\
to 3 & 0.012 & 0.022 & \boxed{{\color{blue}0.030}} & 0.031 & 0.040 & 0.048 & 0.056 & 0.090 & 0.123 & 0.138 & 0.145\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1996;\quad previous event type: 3;\quad holding time: 77 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 450 days & 2 Years & 3 Years & 4 Years \\
to 1 & 0.084 & 0.152 & 0.218 & 0.273 & 0.323 & 0.366 & 0.525 & 0.562 & 0.614 & 0.628 & 0.630 \\
to 2 & 0.036 & 0.063 & 0.090 & 0.113 & 0.133 & 0.150 & 0.219 & 0.236 & 0.266 & 0.277 & 0.281 \\
to 3 & 0.008 & 0.015 & 0.022 & 0.027 & 0.032 & 0.037 & 0.056 &\boxed{{\color{blue}0.062}} & 0.074 & 0.081 & 0.084 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1995;\quad previous event type: 1;\quad holding time: 3100 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 288 days & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 &0.004 & 0.008 & 0.011 & 0.014 & 0.017 & 0.019 & 0.024 & 0.027 & 0.033 & 0.035 & 0.035 \\
to 2 & 0.022 & 0.041 & 0.059 & 0.075 & 0.091 & 0.104 & 0.143 & 0.165 & 0.222 & 0.244 & 0.253\\
to 3 & 0.042 & 0.079 & 0.115 & 0.148 & 0.180 & 0.208 & \boxed{{\color{blue}0.296}} & 0.348 & 0.509 & 0.590 & 0.633 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1994;\quad previous event type: 1;\quad holding time: 2735 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 653 days & 2 Years & 3 Years & 4 Years \\
to 1& 0.005 & 0.010 & 0.014 & 0.018 & 0.021 & 0.023 & 0.034 & 0.041 & 0.041 & 0.043 & 0.043\\
to 2 & 0.024 & 0.043 & 0.062 & 0.080 & 0.096 & 0.110 & 0.174 & 0.225 & 0.233 & 0.256 & 0.265\\
to 3 & 0.041 & 0.076 & 0.112 & 0.144 & 0.175 & 0.204 & 0.341 & \boxed{{\color{blue}0.473}} & 0.497 & 0.575 & 0.617\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1993;\quad previous event type: 1;\quad holding time: 2370 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 1018 days & 3 Years & 4 Years \\
to 1 & 0.007 & 0.013 & 0.019 & 0.024 & 0.028 & 0.031 & 0.045 & 0.055 & 0.056 & 0.057 & 0.057 \\
to 2 & 0.025 & 0.046 & 0.067 & 0.086 & 0.103 & 0.118 & 0.186 & 0.248 & 0.268 & 0.271 & 0.280\\
to 3 & 0.040 & 0.074 & 0.108 & 0.140 & 0.170 & 0.197 & 0.329 & 0.479 & \boxed{{\color{blue}0.542}} & 0.554 & 0.593 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1992;\quad previous event type: 1;\quad holding time: 2005 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 1383 days & 4 Years \\
to 1 & 0.011 & 0.019 & 0.027 & 0.034 & 0.040 & 0.045 & 0.065 & 0.078 & 0.080 & 0.081 & 0.081\\
to 2 & 0.028 & 0.051 & 0.074 & 0.094 & 0.113 & 0.130 & 0.202 & 0.267 & 0.290 & 0.298 & 0.299\\
to 3 & 0.038 & 0.070 & 0.103 & 0.132 & 0.161 & 0.186 & 0.311 & 0.451 & 0.520 & \boxed{{\color{blue}0.551}} & 0.557 \\
\hline
\end{tabular}}
\caption{CSPs as the end of the catalogue shifts back by one-year steps. The numbers in boxes are the probability that the next observed event has occurred at or before the time when it occurred and is of the type that has been observed. The prior was elicited from $\text{MR}_4$.}
\label{table:31.12.92-01}
\end{table}
The examination of the posterior distributions of transition probabilities and of the predictive distributions of the inter-occurrence times can give some insight into the type of energy release and accumulation mechanism. We consider two mechanisms, the time predictable model (TPM) and the slip predictable model (SPM). | 3,999 | 34,990 | en |
train | 0.32.8 | to 2 & 0.022 & 0.041 & 0.059 & 0.075 & 0.091 & 0.104 & 0.143 & 0.165 & 0.222 & 0.244 & 0.253\\
to 3 & 0.042 & 0.079 & 0.115 & 0.148 & 0.180 & 0.208 & \boxed{{\color{blue}0.296}} & 0.348 & 0.509 & 0.590 & 0.633 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1994;\quad previous event type: 1;\quad holding time: 2735 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 653 days & 2 Years & 3 Years & 4 Years \\
to 1& 0.005 & 0.010 & 0.014 & 0.018 & 0.021 & 0.023 & 0.034 & 0.041 & 0.041 & 0.043 & 0.043\\
to 2 & 0.024 & 0.043 & 0.062 & 0.080 & 0.096 & 0.110 & 0.174 & 0.225 & 0.233 & 0.256 & 0.265\\
to 3 & 0.041 & 0.076 & 0.112 & 0.144 & 0.175 & 0.204 & 0.341 & \boxed{{\color{blue}0.473}} & 0.497 & 0.575 & 0.617\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1993;\quad previous event type: 1;\quad holding time: 2370 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 1018 days & 3 Years & 4 Years \\
to 1 & 0.007 & 0.013 & 0.019 & 0.024 & 0.028 & 0.031 & 0.045 & 0.055 & 0.056 & 0.057 & 0.057 \\
to 2 & 0.025 & 0.046 & 0.067 & 0.086 & 0.103 & 0.118 & 0.186 & 0.248 & 0.268 & 0.271 & 0.280\\
to 3 & 0.040 & 0.074 & 0.108 & 0.140 & 0.170 & 0.197 & 0.329 & 0.479 & \boxed{{\color{blue}0.542}} & 0.554 & 0.593 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1992;\quad previous event type: 1;\quad holding time: 2005 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 1383 days & 4 Years \\
to 1 & 0.011 & 0.019 & 0.027 & 0.034 & 0.040 & 0.045 & 0.065 & 0.078 & 0.080 & 0.081 & 0.081\\
to 2 & 0.028 & 0.051 & 0.074 & 0.094 & 0.113 & 0.130 & 0.202 & 0.267 & 0.290 & 0.298 & 0.299\\
to 3 & 0.038 & 0.070 & 0.103 & 0.132 & 0.161 & 0.186 & 0.311 & 0.451 & 0.520 & \boxed{{\color{blue}0.551}} & 0.557 \\
\hline
\end{tabular}}
\caption{CSPs as the end of the catalogue shifts back by one-year steps. The numbers in boxes are the probability that the next observed event has occurred at or before the time when it occurred and is of the type that has been observed. The prior was elicited from $\text{MR}_4$.}
\label{table:31.12.92-01}
\end{table}
The examination of the posterior distributions of transition probabilities and of the predictive distributions of the inter-occurrence times can give some insight into the type of energy release and accumulation mechanism. We consider two mechanisms, the time predictable model (TPM) and the slip predictable model (SPM).
In the TPM, it is assumed that when a maximal energy threshold is reached, some fraction of it (not always the same) is released and an earthquake occurs. The consequence is that the time until the next earthquake increases with the amplitude of the last earthquake. So, the holding time distribution depends on the current event type, but not on the next event type, that is, we expect $F_{ij}(t)=F_{i\cdot}(t)$, $j=1,2,3$. The strength of an event does not depend on the strength of the previous one, because every time the same energy level has to be reached for the event to occur. So we expect $p_{ij}=p_{\cdot j}$, $j=1,2,3$, that is, a transition matrix with equal rows. If this is the case, the CSPs~\eqref{eq:CSP} would simplify as follows,
\begin{equation}\label{eq:CSPTPM}
P_{t_0|\Delta x}^{ij} = \frac{ p_{ij}\left(\bar{F}_{ij}(t_0) - \bar{F}_{ij}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{ik} \bar{F}_{ik}(t_0)}
= \frac{p_{\cdot j} \left(\bar{F}_{i\cdot}(t_0) - \bar{F}_{i\cdot}(t_0+ \Delta x )\right)}{\bar{F}_{i\cdot}(t_0)} \ ,
\end{equation}
so that, under the TPM assumption, given $i$, they are proportional to each other as $j=1,2,3$ for any $\Delta x$, and the ratio
$P_{t_0|\Delta x}^{ij}/P_{t_0|\Delta x}^{ik}$ equals $p_{\cdot j}/p_{\cdot k}$ for any pair $(j,k)$.
In the SPM, after an event, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. The consequence is that the energy of the next earthquake increases with time since the last earthquake. So, the magnitude of an event depends on the length of the holding time, but not on the magnitude of the previous one, because energy always accumulates from the same threshold. In this case again $p_{ij} = p_{\cdot j}$, but $F_{ij}(t) = F_{\cdot j}(t)$, so
\begin{equation}
\label{eq:CSPSPM}
P_{t_0|\Delta x}^{ij} =
\frac{ p_{\cdot j}\left(\bar{F}_{\cdot j}(t_0) - \bar{F}_{\cdot j}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{\cdot k} \bar{F}_{\cdot k}(t_0)} \ .
\end{equation}
Then, under the SPM assumption, CSPs are equal to each other as $i=1,2,3$ for any $\Delta x$, given $j$.
An additional feature that can help discriminate between the TPM and the SPM is the tail of the holding time distribution: for a TPM, the tail of the holding time distribution is thinner \textit{after} a weak earthquake than \textit{after} a strong one; for an SPM, the tail of the holding time is thinner \textit{before} a weak earthquake than \textit{before} a strong one.
In the present case the posterior mean of the third row of $\textbf{p}$, see Table \ref{table:posterior.P}, is clearly different from the other two rows, unlike the empirical transition matrix derived from Table \ref{table:empirical.P}\subref{table:1.empirical.P}, because of the prior information from $\text{MR}_4$. So, with this prior, both the TPM and the SPM are excluded.
On the other hand, things change with the noninformative prior elicited without a learning set.
In this case, we let all the Dirichlet hyperparameters $\g_{ij} $'s be equal to 2. Following Section~ \ref{sec:scarce}, the missing learning set for each string $(i,j)$ is substituted by a unique fictitious observation $\tilde{t}^{ij}_{0.5}$ uniformly distributed over $(1, 5000)$ days, and $m_{ij}$ is set to one; this establishes the prior for $\theta_{ij}$.
The prior of $\alpha_{ij}$ derived from Equation \eqref{pi-3} with $m_{ij}=1$ and $c_{ij}=2$ (taken from Table~\ref{table:hyperparameters}) is
$$
\pi_3(\alpha_{ij}) \propto \left(1-\frac{\alpha_{ij}}{\alpha_{0,ij}}\right) \mathds{1}(\alpha_{0,ij}\le\alpha_{ij}\le\alpha_{1,ij})
$$
with $\alpha_{0,ij}=2/(2+N_{ij})$ (see Table \ref{table:empirical.P}\subref{table:1.empirical.P} for the $N_{ij}$'s) and $\alpha_{1,ij}=10$. Note that on our current sample, the lower limit $\alpha_{0,ij}$ is always smaller than $2/3$.
With this prior specification, the posterior distributions of the rows of the transition matrix do not differ significantly, as seen from Table~\ref{table:lessinformative.posterior.P}, so we can assume $p_{ij}=p_{\cdot j}$ for all indexes $i$ and examine the ratios of CSPs to verify the TPM and the SPM hypotheses.
\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrr}
\hline
& 1 & 2 & 3 \\
\hline
1 & 0.569 (0.046) & 0.271 (0.041)& 0.160 (0.033)\\
2 & 0.568 (0.064)& 0.283 (0.059)& 0.149 (0.046)\\
3 & 0.500 (0.085)& 0.323 (0.078)& 0.177 (0.065)\\
\hline
\end{tabular}
\end{center}
\caption{Posterior means (with standard deviations) of the transition matrix $\bp$ with the noninformative prior.}
\label{table:lessinformative.posterior.P}
\end{table}
Figures~\ref{figure:improper_ratio_TPM} and \ref{figure:improper_ratio_SPM} display the posterior means of the ratios of the CSPs as a function of $\Delta x$ for $t_0=0$, with the noninformative prior. For the TPM the generic ratio of two CSPs indexed by $(i,j)$ and $(i,k)$ should be approximately constant and close to $p_{\cdot j}/p_{\cdot k}$, where the $p_{\cdot j}$ represents the common values of the entries in the $j$-th column of $\bp$, under the TPM. The horizontal lines in Figure~\ref{figure:improper_ratio_TPM} are the posterior expectations of $p_{ij}/p_{ik}$, which would estimate $p_{\cdot j}/p_{\cdot k}$ if the TPM assumption were true. For the SPM, the ratio of CPSs, now indexed by $(i,j)$ and $(k,j)$, should be close to one. The plots indicate that it is not so, therefore neither the TPM nor the SPM are supported by the data.
As for the TPM, this finding is confirmed by the examination of the posterior probabilities that $\a_{ij}<\a_{ik}$ and $\theta_{ij}<\theta_{ik}$, for any given $i$ and $j\ne k$: $\Pr(\a_{ij}<\a_{ik}|\bj, \bx, u_T)$ is 0.55 for string $(2,1)$ versus string $(2,3)$ and 0.51 for $(3,1)$ versus $(3,2)$, but it is either larger than 0.75 or smaller than 0.35 for all the other strings; $\Pr(\theta_{ij}<\theta_{ik}|\bj, \bx, u_T)$ is 0.61 for $(1,2)$ versus $(1,3)$ and is 0.53 for $(3,1)$ versus $(3,2)$, but it is lower than 0.39 for all the other strings. As for the SPM, we have examined $\Pr(\a_{ij}<\a_{kj}|\bj, \bx, u_T)$ and $\Pr(\theta_{ij}<\theta_{kj}|\bj, \bx, u_T)$ for any $j$ and $i\ne k$: $\Pr(\a_{ij}<\a_{kj}|\bj, \bx, u_T)$ is 0.51 for $(2,2)$ versus $(3,2)$, but is either larger than 0.75 or smaller than 0.35 for all the other strings; $\Pr(\theta_{ij}<\theta_{kj}|\bj, \bx, u_T)$ is between 0.44 and 0.63 for three comparisons but is either larger than 0.73 or smaller than 0.30 for the remaining ones.
\begin{figure}
\caption{Checking the TPM: posterior means of ratios of CSPs $P_{0|\Delta x}
\label{figure:improper_ratio_TPM}
\end{figure}
\begin{figure}
\caption{Checking the SPM: posterior means of ratios of CSPs $P_{0|\Delta x}
\label{figure:improper_ratio_SPM}
\end{figure} | 3,853 | 34,990 | en |
train | 0.32.9 | \section{Concluding remarks}
\label{sec:6}
We have presented a complete Bayesian methodology for the inference on semi-Markov processes, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation.
In particular, we have examined in detail the elicitation of the joint prior density of the shape and scale parameters of the Weibull-distributed holding times (conditional on the transition between two given states), deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning data'' and moment existence conditions. This framework has been applied to the analysis of seismic data, but it can be adopted for inference on any system for which a Markov Renewal process is plausible. A possible and not-yet explored application is the modelling of voltage sags (or voltage dips) in power engineering: the state space would be formed by different classes of voltage, starting from voltage around its nominal value, down to progressively deeper sags. In the engineering literature, the dynamic aspect of this problem is in fact disregarded, while it could help bring additional insight into this phenomenon.
With regard to the seismic data analysis, other uses of our model can be envisaged. The model can be applied to areas with a less complex tectonics, such as Turkey, by replicating for example Alvarez's analysis. Outliers, such as those appearing in Figure \ref{figure:caterpillar_times}, could point at events whose assignment to a specific seismogenic source should be re-discussed. The analysis of earthquake occurrence can support decision making related to the risk of future events. We have not examined this issue here, but a methodology is outlined by \cite{Cano}.
A final note concerns the more recent Italian seismic catalogue \cite{CPTI11}, including events up to the end of 2006. Every new release of the catalogue involves numerous changes in the parameterization of earthquakes; as the DISS event classification by macroregion is not yet available for events in this catalogue we cannot use this more recent source of data.
\appendix | 497 | 34,990 | en |
train | 0.32.10 | \section{Gibbs sampling}
\label{sec:gibbsampling}
Here we derive the full conditional distributions involved in the Gibbs sampling and give indications on its JAGS implementation. | 46 | 34,990 | en |
train | 0.32.11 | \subsection{Full conditional distributions}
Let the last holding time be censored i.e. $u_T>0$. Hence,
in order to obtain some simple full conditional distributions and then an efficient Gibbs sampling, we introduce the auxiliary variable $j_{\tau+1}$
which represents the unobserved state following the last visited state $j_\t$. Moreover, let
$\bt_{\bq} = (t^{ij}_{q_{ij}}, \;\; i,j\in E)$. Each hyperparameter $t^{ij}_{q_{ij}}$ may be either a known constant or $t^{ij}_{q_{ij}}$ is uniformly distributed over an interval $(t_1, t_2)$. Moreover, all of them are independent of each other.
Thus the state space of the Gibbs sampler is $(\bp, \ba, \bth,j_{\t+1}, \bt_{\bq})$ and the following full likelihood derived from \eqref{eq:likelihood-param}:
\begin{multline}
\label{eq:full-likelihood-param}
L(\bj, \bx, u_T , j_{\t+1}|\bp, \ba, \bth) =\prod_{i,k\in E}p_{ik}^{N_{ik}}\times \\
\times \prod_{i,k\in E}\left[\a_{ik}^{N_{ik}}\frac{1}{\theta_{ik}^{\a_{ik}N_{ik}}}\left(\prod_{\rho=1}^{N_{ik}}x^{\r}_{ik}\right)^{\a_{ik}-1} \times
\exp\left\{-\frac{1}{\theta_{ik}^{\a_{ik}}}\sum_{\rho =1}^{N_{ik}}(x^{\r}_{ik})^{\a_{ik}}\right\}\right]\times \\
\times \left(p_{j_{\t} j_{\t+1}}
\exp\left\{-\left(\frac{u_T}{\theta_{j_{\t} j_{\t+1}}}\right)^{\a_{j_{\t} j_{\t+1}}}\right\}\right)\
\end{multline}
is multiplied by the prior and used to determine the full conditionals.
For every $i$ and $j$ let
\begin{align*}
\bp_{(-i)} & = \mbox{the transition matrix $\bp$ without the $i$-th row} , \\
\ba_{(-ij)} & = (\a_{hk}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) \ , \\
\bth_{(-ij)} & = (\th_{hk}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) \ , \\
\bt_{{\bq}(-ij)} & = (t^{hk}_{q_{hk}}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) , \\
\tilde{N}_{ij} & = {N}_{ij} + \mathds{1}{\left((j_{\t},j_{\t+1})=(i,j)\right)}, \quad
\tilde{m_{\alpha}thbf N}_{i} =\left(\tilde{N}_{ij}, \;\; j=1,\ldots, s \right),\\
\tilde{M}_{ij}(\a_{ij}) & = \sum_{\rho=1}^{{N}_{ij}}(x_{ij}^{\rho})^{\a_{ij}} + \ u_{T}^{\a_{ij}}\mathds{1}( (j_{\t},j_{\t+1})=(i,j)), \\
{\C}_{ij} & = \prod_{\rho=1}^{{N}_{ij}}x_{ij}^{\rho} \ .
\end{align*}
The following result on the full conditional distributions of the Gibbs sampling holds.
\begin{proposition}
\label{prop:A.1}
Let the prior on $(\bp,\ \ba, \ \bth, \ \bt_{\bq})$ be the following
\begin{enumerate}[i)]
\item $\bp$ is independent of $\ba$ and $\bth$ and the rows of $\bp$ are $s$ independent vectors with Dirichlet distribution with parameters $\bg_1, \cdots ,\bg_s$ and total mass $c_1, \cdots c_s$, respectively,
\item the $\th_{ij}$'s, given the $\a_{ij}$'s and the $t^{ij}_{q_{ij}}$'s, are independent with $\th_{ij}|\a_{ij} \sim {m_{\alpha}thcal GIG}(m_{ij}, b_{ij}(t^{ij}_{q_{ij}},\a_{ij}), \a_{ij})$, where
\begin{equation*}
b_{ij}(t^{ij}_{q_{ij}},\a_{ij}) = \left(t^{ij}_{q_{ij}}\right)^{\a_{ij}}[(1-q_{ij})^{-1/m_{ij}}-1]^{-1} \ ,
\end{equation*}
and $t^{ij}_{q_{ij}}$ is either a known constant or $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$,
\item $\pi_{3,ij}(\a_{ij})\propto \a_{ij}^{m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}\exp\{-m_{ij}d_{ij}\a_{ij}\}\mathds{1}(\a_{ij}\in {I_{ij}})$,
$m_{ij}>0, c_{ij}>0$, $\a_{0,ij}>0 \mbox{ and } d_{ij}\ge0$ where
$$
I_{ij} =
\begin{cases}
(\a_{0,ij}, \a_{1,ij}) & \mbox{ if } d_{ij} = 0\\
(\a_{0,ij}, \infty) & \mbox{ if } d_{ij} > 0 \ .
\end{cases}
$$
\end{enumerate}
Then
\begin{enumerate}[$a)$]
\item the conditional distribution of $\bp_i$, given $\bj, \ \bx, \ u_T, \ j_{\t+1},\ \bp_{(-i)}, \ \ba, \ \bth$ and $\bt_{\bq}$ is a Dirichlet distribution with parameter $\tilde{m_{\alpha}thbf N}_{i}+\bg_i$;
\item the conditional distribution of $\th_{ij}^{\a_{ij}}$, given $\bj, \ \bx , \ u_T, \ j_{\t+1}, \ \bp , \ \ba, \ \bth_{(-ij)}$ and $\bt_{\bq}$ is an inverse Gamma distribution with shape $m_{ij}+{N}_{ij}$ and rate $b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij})$;
\item the conditional density of $\a_{ij}$, given $\bj , \ \bx, \ u_T, \ j_{\t+1}, \ \bp, \ \ba_{(-ij)}, \ \bth$ and $\bt_\bq$ is proportional to
\begin{multline}
\label{eq:pi3_posterior}
\a_{ij}^{N_{ij}+1+m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}
\times
\exp\left\{ - \left(m_{ij}d_{ij} -\log \frac{\C_{ij} (t^{ij}_{q_{ij}})^{m_{ij}}}{\th_{ij}^{N_{ij} + m_{ij}}}\right)\a_{ij}\right\}
\times \\
\times \exp\left\{- \frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij}) }{\th_{ij}^{\a_{ij}}}\right\} \mathds{1}(\a_{ij}\in {I_{ij}})\ ,
\end{multline}
and it is log-concave if $c_{ij}\ge 1$;
\item the conditional density of the unseen state $J_{\t+1}$, given $\bj, \ \bx, \ u_T, \ \bp, \ \ba,\bth$ and $\bt_{\bq}$, is
\begin{equation*}
\frac{p_{j_{\t}j}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}j}}\right)^{\a_{j_{\t}j}}\right\}}{\sum_{k\in E}
p_{j_{\t}k}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}k}}\right)^{\a_{j_{\t}k}}\right\}} \, ;
\end{equation*}
\item if $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$, then its conditional distribution given $\bj, \ \bx, \ u_T, \ \bp, \ \ba, \ \bth$ and $\bt_{{\bq}(-ij)}$ is a doubly-truncated at $(t_1, t_2)$ generalized Gamma with parameters
$a= \th_{ij}[(1-q_{ij})^{-1/m_{ij}}-1]^{1 /\a_{ij}}$, $d=\a_{ij}m_{ij}+1$ and $p=\alpha_{ij}$, i.e.
$$
\pi(t_{q_{ij}}^{ij} | \bj, \bx, u_T, \bp, \ba, \bth, \bt_{{\bq}(-ij)}) =
\frac{p/a^d}{\Gamma(d/p)} \left(t_{q_{ij}}^{ij}\right)^{d-1} \exp\left\{-\left(t_{q_{ij}}^{ij} / a \right)^p \right\} \mathds{1}(t_1 <t^{ij}_{q_{ij}}< t_2) .
$$
\end{enumerate}
\end{proposition}
\begin{proof}
As the row $\bp_{i}$ is independent of $(\bp_{(-i)},\ba,\bth, \bt_{\bq})$, conditionally on the data and $j_{\t+1}$, then
\begin{equation*}
\pi(\bp_i|\bj, \bx , u_T, \ j_{\t+1} ,\bp_{(-i)}, \ba,\bth,\bt_{\bq}) \propto L(\bj, \bx, u_T, \ j_{\t+1}| \bp, \ba,\bth)
\times \pi_{1,i}(\bp_i) \propto \prod_{j\in E} p_{ij}^{\tilde{N}_{i,j}} \times \prod_{j\in E} p_{ij}^{\gamma_{ij}-1} ,
\end{equation*}
where $\pi_{1,i}$ denotes the Dirichlet prior of $\bp_i$. Hence point $a)$ of Proposition \ref{prop:A.1} follows.
As regards the full conditional distribution of $\theta_{ij}$,
we have
\begin{align*}
\pi(\th_{ij} & \ |\ \bj, \bx , u_T, \ j_{\t+1}, \bp, \ba,\bth_{(-ij)},\bt_{\bq})
\propto L(\bj, \bx, u_T, \ j_{\t+1} \ |\bp, \ba,\bth) \times \pi_{2,ij}(\th_{ij}|\alpha_{ij}, t^{ij}_{q_{ij}}) \\
& \propto
\prod_{i,k\in E} \left[\a_{ik}^{N_{ik}}\frac{\C_{ik}^{\a_{ik}-1}}{\theta_{ik}^{\a_{ik}N_{ik}}}
\times\exp\left\{-\frac{\tilde{M}_{ik}(\a_{ik})}{\theta_{ik}^{\a_{ik}}}\right\} \right]
\times \exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}\th_{ij}^{-[1+\a_{ij}m_{ij}]}\\
& \propto \; \; \th_{ij}^{-[1+\a_{ij}(m_{ij}+N_{ij})]}
\exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}.
\end{align*}
As one can see, the last function is the kernel of an inverse Gamma distribution with parameters $m_{ij}+N_{ij}$ and $b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{i,j}(\a_{ij})$ and point $b)$ follows. \newline
A similar reasoning yields a full conditional distribution for $\a_{ij}$
proportional to \eqref{eq:pi3_posterior}.
Furthermore, concerning its log-concavity, notice that the function in \eqref{eq:pi3_posterior} can be written as the product of the following four log-concave functions:
\begin{align*}
& \a_{ij}^{N_{ij}+m_{ij}} , \ \Bigl(1-\frac{\a_{ij}}{\a_{0,ij}}\Bigr)^{c_{ij}-1} , \
\exp\Bigl\{ - \bigl(m_{ij}d_{ij} -\log \frac{\C_{ij} (t^{ij}_{q_{ij}})^{m_{ij}}}{\th_{ij}^{N_{ij} + m_{ij}}}\bigr)\a_{ij}\Bigr\} , \
\exp\Bigl\{- \frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij}) }{\th_{ij}^{\a_{ij}}}\Bigr\}
\end{align*}
In particular,
the second function is log-concave for $c_{ij} \ge 1$ and
the last term is a product of log-concave functions of the kind $\a_{ij} m_{\alpha}psto \exp\{-z^{\a_{ij}}\}$.
Hence the log-concavity follows from the property that the product of log-concave functions is log-concave too. | 3,533 | 34,990 | en |
train | 0.32.12 | \frac{p_{j_{\t}j}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}j}}\right)^{\a_{j_{\t}j}}\right\}}{\sum_{k\in E}
p_{j_{\t}k}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}k}}\right)^{\a_{j_{\t}k}}\right\}} \, ;
\end{equation*}
\item if $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$, then its conditional distribution given $\bj, \ \bx, \ u_T, \ \bp, \ \ba, \ \bth$ and $\bt_{{\bq}(-ij)}$ is a doubly-truncated at $(t_1, t_2)$ generalized Gamma with parameters
$a= \th_{ij}[(1-q_{ij})^{-1/m_{ij}}-1]^{1 /\a_{ij}}$, $d=\a_{ij}m_{ij}+1$ and $p=\alpha_{ij}$, i.e.
$$
\pi(t_{q_{ij}}^{ij} | \bj, \bx, u_T, \bp, \ba, \bth, \bt_{{\bq}(-ij)}) =
\frac{p/a^d}{\Gamma(d/p)} \left(t_{q_{ij}}^{ij}\right)^{d-1} \exp\left\{-\left(t_{q_{ij}}^{ij} / a \right)^p \right\} \mathds{1}(t_1 <t^{ij}_{q_{ij}}< t_2) .
$$
\end{enumerate}
\end{proposition}
\begin{proof}
As the row $\bp_{i}$ is independent of $(\bp_{(-i)},\ba,\bth, \bt_{\bq})$, conditionally on the data and $j_{\t+1}$, then
\begin{equation*}
\pi(\bp_i|\bj, \bx , u_T, \ j_{\t+1} ,\bp_{(-i)}, \ba,\bth,\bt_{\bq}) \propto L(\bj, \bx, u_T, \ j_{\t+1}| \bp, \ba,\bth)
\times \pi_{1,i}(\bp_i) \propto \prod_{j\in E} p_{ij}^{\tilde{N}_{i,j}} \times \prod_{j\in E} p_{ij}^{\gamma_{ij}-1} ,
\end{equation*}
where $\pi_{1,i}$ denotes the Dirichlet prior of $\bp_i$. Hence point $a)$ of Proposition \ref{prop:A.1} follows.
As regards the full conditional distribution of $\theta_{ij}$,
we have
\begin{align*}
\pi(\th_{ij} & \ |\ \bj, \bx , u_T, \ j_{\t+1}, \bp, \ba,\bth_{(-ij)},\bt_{\bq})
\propto L(\bj, \bx, u_T, \ j_{\t+1} \ |\bp, \ba,\bth) \times \pi_{2,ij}(\th_{ij}|\alpha_{ij}, t^{ij}_{q_{ij}}) \\
& \propto
\prod_{i,k\in E} \left[\a_{ik}^{N_{ik}}\frac{\C_{ik}^{\a_{ik}-1}}{\theta_{ik}^{\a_{ik}N_{ik}}}
\times\exp\left\{-\frac{\tilde{M}_{ik}(\a_{ik})}{\theta_{ik}^{\a_{ik}}}\right\} \right]
\times \exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}\th_{ij}^{-[1+\a_{ij}m_{ij}]}\\
& \propto \; \; \th_{ij}^{-[1+\a_{ij}(m_{ij}+N_{ij})]}
\exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}.
\end{align*}
As one can see, the last function is the kernel of an inverse Gamma distribution with parameters $m_{ij}+N_{ij}$ and $b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{i,j}(\a_{ij})$ and point $b)$ follows. \newline
A similar reasoning yields a full conditional distribution for $\a_{ij}$
proportional to \eqref{eq:pi3_posterior}.
Furthermore, concerning its log-concavity, notice that the function in \eqref{eq:pi3_posterior} can be written as the product of the following four log-concave functions:
\begin{align*}
& \a_{ij}^{N_{ij}+m_{ij}} , \ \Bigl(1-\frac{\a_{ij}}{\a_{0,ij}}\Bigr)^{c_{ij}-1} , \
\exp\Bigl\{ - \bigl(m_{ij}d_{ij} -\log \frac{\C_{ij} (t^{ij}_{q_{ij}})^{m_{ij}}}{\th_{ij}^{N_{ij} + m_{ij}}}\bigr)\a_{ij}\Bigr\} , \
\exp\Bigl\{- \frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij}) }{\th_{ij}^{\a_{ij}}}\Bigr\}
\end{align*}
In particular,
the second function is log-concave for $c_{ij} \ge 1$ and
the last term is a product of log-concave functions of the kind $\a_{ij} m_{\alpha}psto \exp\{-z^{\a_{ij}}\}$.
Hence the log-concavity follows from the property that the product of log-concave functions is log-concave too.
Regarding point $d)$, it is enough to observe that Equations~\eqref{eq:MRP_n} and \eqref{eq:full-likelihood-param} imply the following:
\begin{align*}
P(J_{\t+1}=j |\ \bj, \bx, u_T, \bp , \ba, \bth ) & =
\frac{P(J_{\t+1}=j, \ X_{\tau +1} >u_T | \ \bj, \bx,\bp , \ba, \bth)}{\sum_{k\in E}P(J_{\t+1}=k, \ X_{\tau +1} >u_T | \ \bj, \bx,\bp , \ba, \bth)} \\
& =\frac{p_{j_{\t}j}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}j}}\right)^{\a_{j_{\t}j}}\right\}}{\sum_{k\in E}
p_{j_{\t}k}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}k}}\right)^{\a_{j_{\t}k}}\right\}} \ .
\end{align*}
Finally, if $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$, then
\begin{align*}
\pi(t_{q_{ij}}^{ij} &| \ \bj, \bx, u_T, \bp, \ba, \bth, \bt_{{\bq}(-ij)}) \propto
\pi_2(\th_{ij}|\a_{ij},t^{ij}_{q_{ij}})\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2) \\
&\propto b^{m_{ij}}(t^{ij}_{q_{ij}},\a_{ij})\exp\left\{-\frac{b(t^{ij}_{q_{ij}},\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2) \\
&\propto \left(t_{q_{ij}}^{ij}\right)^{\a_{ij}m_{ij}} \exp\left\{-\left(\frac{t_{q_{ij}}^{ij}}{\th_{ij}[(1-q_{ij})^{-1/m_{ij}}-1]^{1 /\a_{ij}}}\right)^{\a_{ij}} \right\}
\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2)
\end{align*}
The last equation is the kernel of a generalized Gamma, doubly-truncated at $(t_1, t_2)$, as introduced in \cite{Stacy}, so point $e)$ follows.
\end{proof} | 2,193 | 34,990 | en |
train | 0.32.13 | \subsection{JAGS implementation} Proposition~\ref{prop:A.1} implies that JAGS should be able to run an exact Gibbs sampler.
The model description we have adopted in the JAGS language is based on the full likelihood \eqref{eq:full-likelihood-param}.
It is not important that the model description matches the actual model which generated the data as long as the full conditional distributions, which are determined by the joint distribution of the data and the parameters, remain unchanged. In detail, we consider the following joint distribution:
$$
L_1(\bj,\bx,u_T|j_{\tau+1},\bp,\ba,\bth) \pi(j_{\tau+1}|\bp_{j_\tau}) \pi(\bp,\ba,\bth)
$$
where $\pi(\bp,\ba,\bth)$ is the joint prior as derived in Section~\ref{sec:3} using Equations \eqref{eq:dirich}, \eqref{pi-2} and \eqref{pi-3} and
\begin{align*}
\pi(j_{\tau+1}|\bp_{j_\tau}) & = p_{j_\tau j_{\tau+1}}\\
L_1(\bj,\bx,u_T|j_{\tau+1},\bp,\ba,\bth) & = L(\bj,\bx,u_T,j_{\tau+1}|\bp,\ba,\bth)/p_{j_\tau j_{\tau+1}}
\end{align*}
The factors of the likelihood $L_1$, to be extracted from Equation \eqref{eq:full-likelihood-param}, are modelled in JAGS as follows.
For any value of $i$, the factor
$$
\prod_{k\in E} p_{ik}^{N_{ik}}
$$
is contributed by a multinomial likelihood with probability vector $\bp_i$ and $\sum_k N_{ik}$ trials.
The factors in square brackets in \eqref{eq:full-likelihood-param} are contributed by the uncensored Weibull holding times for every string $(i,k)$ and are obtained in JAGS as Weibull densities with parameters $\alpha_{ik}$ and $\theta_{ik}$, using a ``for'' loop sweeping the strings. The last factor, which accounts for the censored holding time $u_T$, is handled by a special instruction, by which $u_T$ is first declared to be a right censored time with upper censoring point $T-t_\tau$, and then is assigned a Weibull distribution with parameters $\alpha_{j_\tau j_{\tau+1}}$ and $\theta_{j_\tau j_{\tau+1}}$.
The factor $\pi(j_{\tau+1}|\bp_{j_\tau}) \pi(\bp,\ba,\bth)$, representing the prior associated with $L_1$, is handled as follows. The additional prior on $j_{\tau+1}$ is a discrete distribution on the integers 1, 2, 3, with probabilities taken from the row of $\bp$ indexed by $j_\tau$. Every row $\bp_i$ of $\bp$ is assigned a Dirichled distribution directly. Whenever $m_{ik}\ge 2$, as $c_{ik}=m_{ik}$, the shape parameters $\alpha_{ik}$ have a shifted Gamma prior, see Equation \eqref{pi-3}, obtainable by defining in JAGS a new non-shifted Gamma variable with shape $c_{ik}$ and rate $m_{ik}d_{ik}$, which, after summing the shift, is assigned to $\alpha_{ik}$; for the value of $d_{ik}$ see Equation \eqref{eq:d_ij}. The generalized Gamma for $\theta_{ik}$ is defined conditionally on $\alpha_{ik}$: first a Gamma prior with shape $m_{ij}$ and rate $\hat{b}_{ik}(\alpha_{ik})$ is assigned to a new random variable $a_{ik}$ and then $a_{ik}^{-1/\a_{ik}}$ is assigned to $\theta_{ik}$.
In case there is either just one observation or no learning dataset $\by_m$, some special instructions in the JAGS code are needed.
In particular, if there is no learning dataset, then the missing learning dataset is substituted, for every string $(i,k)$, by the fictitious observation $\tilde{t}_{ik}$ drawn from a uniform distribution over $(1,5000)$ days (so $m_{ik}=1$ for all strings). Then, the priors of the $\theta_{ik}$ retain the same form, whereas the priors of the $\alpha_{ik}$'s, derived from Equation \eqref{pi-3} with $m_{ik}=1$ and $c=2$ (a value taken from Table~\ref{table:hyperparameters}), are
$$
\pi_3(\alpha_{ik}) \propto \left(1-\frac{\alpha_{0,ik}}{\alpha_{ik}}\right) \mathds{1}(\alpha_{0,ik}\le\alpha_{ik}\le\alpha_{1,ik})
$$
with assigned $\alpha_{0,ik}$ and $\alpha_{1,ik}$.
This latter distribution is coded using the so-called zeros trick: a fictitious zero observation from a Poisson distribution with mean $\phi_{ik}=-\ln \pi_3(\alpha_{ik})$ is introduced; then a uniform prior over $[\a_{0,ik},\a_{1,ik}]$ is assigned to $\alpha_{ik}$. The effect on the formula of the joint distribution is that the likelihood $L_1$ gets multiplied by the factor $\exp(-\phi_{ik})$, contributed by the zero observation; the multiplication by the uniform density gives back the correct factor accounting for the prior of $\alpha_{ik}$.
\section*{\large{Acknowledgments}}
We are grateful to Renata Rotondi for providing us with data and the map of Italy along with her very helpful comments. We are solely responsible for any remaining inaccuracy.
\end{document} | 1,372 | 34,990 | en |
train | 0.33.0 | \betagin{document}
\title[Identities for a parametric Weyl algebra]{Identities for a parametric Weyl algebra over a ring}
\thanks{The first author was supported by CNPq 313358/2017-6, FAEPEX 2054/19 and FAEPEX 2655/19}
\author{Artem Lopatin}
\address{Artem Lopatin\\
State University of Campinas, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil}
\email{[email protected] (Artem Lopatin)}
\author{Carlos Arturo Rodriguez Palma}
\address{Carlos Arturo Rodriguez Palma\\
State University of Campinas, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil; Industrial University of Santander, Cl.~9 \#Cra 27, Ciudad Universit\'aria,
Bucaramanga, Santander, Colombia}
\email{[email protected] (Carlos Arturo Rodriguez Palma)}
\betagin{abstract}
In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra $\mathsf{A}_h$ generated by elements $x,y,$ which satisfy the relation $yx-xy=h$ for some $0\neq h\in \mathbb{F}[x]$. We generalize this construction to $\mathsf{A}_h(\mathsf{B})$ by working over the fixed $\mathbb{F}$-algebra $\mathsf{B}$ instead of $\mathbb{F}$. We describe the polynomial identities for $\mathsf{A}_h(\mathsf{B})$ over the infinite field $\mathbb{F}$ in case $h\in\mathsf{B}[x]$ satisfies certain restrictions.
\noindent{\bf Keywords: } polynomial identities, matrix identities, Weyl algebra, Ore extensions, positive characteristic.
\noindent{\bf 2020 MSC: } 16R10; 16S32.
\end{abstract}
\maketitle | 554 | 28,609 | en |
train | 0.33.1 | \section{Introduction}
Assume that $\mathbb{F}$ is a field of arbitrary characteristic $p=\mathop{\rm char}\mathbb{F}\geq0$. All vector spaces and algebras are over $\mathbb{F}$ and all algebras are associative, unless other\-wise is stated. For the fixed $\mathbb{F}$-algebra $\mathsf{B}$ with unity we write $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ for the $\mathbb{F}$-algebra of non-commutative $\mathsf{B}$-polynomials in variables $x_1,\ldots,x_m$, i.e., $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ is a free left (and a free right) $\mathsf{B}$-module with the basis given by the set of all non-commutative monomials in $x_1,\ldots,x_m$, where we assume that $\beta x_i = x_i \beta$ for all $\beta\in \mathsf{B}$ and $1\leq i\leq m$. The unity $1$ of $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ corresponds to the empty monomial. In case the variables are $x_1,x_2,\dotsc$ the algebra of non-commutative $\mathsf{B}$-polynomials is denoted by $\mathsf{B}\lambdangle X\mathsf{R}A$. Similarly, we define the algebra of commutative $\mathsf{B}$-polynomials $\mathsf{B}[x_1,\ldots,x_m]$ as a free left (and a free right) $\mathsf{B}$-module with the basis given by the set of all monomials in $x_1^{i_1}\cdots x_m^{i_m}$ with $i_1,\ldots,i_m\geq0$, where we assume that $\beta x_i = x_i \beta$ and $x_i x_j=x_j x_i$ for all $\beta\in \mathsf{B}$ and $1\leq i,j\leq m$. Note that $\mathsf{B}\lambdangle x\mathsf{R}A = \mathsf{B}[x]$.
\subsection{Parametric Weyl algebra $\mathsf{A}_h(\mathsf{B})$ as the Ore extension}
We study the polynomial identities for the following family of infinite-dimensional unital algebras $\mathsf{A}_h(\mathsf{B})$, which are parametrized by a polynomial $h$ from the center of $\mathsf{B}[x]$:
\betagin{defin}\lambdabel{Def Ah}
For $h\in Z(\mathsf{B})[x]$, the {\it parametric Weyl algebra} $\mathsf{A}_h(\mathsf{B})$ {\it over the ring} $\mathsf{B}$ is the unital associative algebra over $\mathbb{F}$ generated by $\mathsf{B}$ and letters $x$, $y$ commuting with $\mathsf{B}$ subject to the defining relation $yx=xy+h$ (equivalently, $[y,x]=h$, where $[y,x]=yx-xy$), i.e., $$\mathsf{A}_h(\mathsf{B})=\mathsf{B}\lambdangle x,y\rangle/\id{yx-xy-h}.$$
\end{defin}
For short, we denote $\mathsf{A}_h=\mathsf{A}_h(\mathbb{F})$. The partial cases of the given construction are the Weyl algebra $\mathsf{A}_1$, the polynomial algebra $\mathsf{A}_0=\mathbb{F}[x,y]$, and the universal enveloping algebra $\mathsf{A}_x$ of the two-dimensional nonabelian Lie algebra. For $h\in \mathbb{F}[x]$, the following isomorphism of $\mathbb{F}$-algebras holds:
\betagin{eq}\lambdabel{eq_tensor}
\mathsf{A}_h(\mathsf{B}) \sigmameq \mathsf{B} \otimes_{\mathbb{F}} \mathsf{A}_h.
\end{eq}
Note that in general the isomorphism~\mathsf{R}ef{eq_tensor} does not hold because $A_h$ is not well-defined in case $h\not\in\mathbb{F}[x]$. Given a polynomial $f= \eta_d x^d + \eta_{d-1} x^{d-1} + \cdots + \eta_0$ of $\mathsf{B}[x]$ with $d\geq0$, we say that $\eta_d$ is the {\it leading coefficient} of $f$ and the product $\eta_d x^d$ is the {\it leading term} of $f$.
The algebra $\mathsf{A}_h$ was introduced and studied by Benkart, Lopes, Ondrus~\cite{Benkart_Lopes_Ondrus_I, Benkart_Lopes_Ondrus_II, Benkart_Lopes_Ondrus_III} as a natural object in the theory of Ore extensions. In particular, they determined automorphisms of $\mathsf{A}_h$ over an arbitrary field $\mathbb{F}$ and the invariants of $\mathsf{A}_h$ under the automorphisms, completely described the simple modules and derivations of $\mathsf{A}_h$ over any field. Then Lopes and Solotar~\cite{Lopes_Solotar_2019} described the Hochschild cohomology ${\rm HH}^{\bullet}(\mathsf{A}_h)$ over a field of arbitrary characteristic. Over an algebraically closed field of zero characteristic simple $\mathsf{A}_h$-modules were independently classified by Bavula~\cite{Bavula_2020}. In recent preprints~\cite{Bavula_2021_zero},~\cite{Bavula_2021_prime} Bavula continued the study of the automorphism group of $\mathsf{A}_h$.
Let us recall that an Ore extension of $R$ (or, equivalently, a skew polynomial ring over $R$) $A=\mathsf{R}[y,\sigmagma,\deltalta]$ is given by a unital associative (not necessarily commutative) algebra $\mathsf{R}$ over a field $\mathbb{F}$, an $\mathbb{F}$-algebra endomorphism $\sigmagma:\mathsf{R}\rightarrow \mathsf{R}$, and a $\sigmagma$-derivation $\deltalta:\mathsf{R}\rightarrow \mathsf{R}$, i.e., $\deltalta$ is $\mathbb{F}$-linear map and $\deltalta(ab)=\deltalta(a)b+\sigmagma(a)\deltalta(b)$ for all $a,b\in \mathsf{R}$. Then $A=\mathsf{R}[y,\sigmagma,\deltalta]$ is the unital algebra generated by $y$ over $\mathsf{R}$ subject to the relation $$ya=\sigmagma(a)y+\deltalta(a) \quad \text{ for all } a\in \mathsf{R}.$$
Assume that $\mathsf{R}=\mathsf{B}[x]$, $\sigmagma={\rm id}_{\mathsf{R}}$ is the identity automorphism on $\mathsf{R}$, and $\deltalta:\mathsf{R}\rightarrow \mathsf{R}$ is given by $\deltalta(a)=a'h$ for all $a\in \mathsf{R}$, where $a'$ stands for the usual derivative of a $\mathsf{B}$-polynomial $a$ with respect to the variable $x$. Since $h\in Z(\mathsf{R})$, $\delta$ is a derivation of $\mathsf{R}$. Using the linearity of derivative and induction on the degree of $a\in\mathsf{B}[x]$ it is easy to see that
\betagin{eq}
[y,a]=a'h \text{ holds in }\mathsf{A}_h(\mathsf{B}) \text{ for all }a\in \mathsf{B}[x].
\end{eq}
\noindent{}Hence $\mathsf{A}_h(\mathsf{B})=\mathsf{R}[y,\sigmagma,\deltalta]$ is an Ore extension. The following lemma is a corollary of Observation 2.1 from~\cite{AVV} proven by Awami, Van den Bergh and Van Oystaeyen (see also Proposition 3.2 of \cite{AD2} and Lemma 2.2 of \cite{Benkart_Lopes_Ondrus_I}).
\betagin{lemma} Assume that $A=\mathsf{R}[y,\sigma,\delta]$ is an Ore extension of $\mathsf{R}=\mathbb{F}[x]$, where $\sigma$ is an automorphism of $\mathsf{R}$. Then $A$ is isomorphic to one of the following algebras:
\betagin{enumerate}
\item[$\bullet$] a quantum plane, i.e., $A\sigmameq\FF\lambdangle x,y\rangle/\id{yx-q xy}$ for some $q\in\mathbb{F}^{*}=\mathbb{F}\backslash \{0\}$;
\item[$\bullet$] a quantized Weyl algebra, i.e., $A\sigmameq\FF\lambdangle x,y\rangle/\id{yx-q xy-1}$ for some $q\in\mathbb{F}^{*}$;
\item[$\bullet$] an algebra $\mathsf{A}_h$ for some $h\in\mathbb{F}[x]$.
\end{enumerate}
\end{lemma}
\noindent{}Note that by Theorem 9.3 of~\cite{Benkart_Lopes_Ondrus_I} the algebra $\mathsf{A}_h$ is not a generalized Weyl algebra over $\mathbb{F}[x]$ in the sense of Bavula~\cite{Bavula_1992} in case $h\not\in\mathbb{F}$.
Since the algebra of $\mathsf{B}$-polynomials $\mathsf{B}[x,y]$ is well studied, in what follows we assume that $h$ is non-zero. Moreover, we assume that the following restriction holds:
\betagin{conv}\lambdabel{conv1} The leading coefficient of $h\in Z(\mathsf{B})[x]$ is not a zero divisor.
\end{conv}
\subsection{Polynomial identities}
A polynomial identity for a unital $\mathbb{F}$-algebra $\mathcal{A}$ is an element $f(x_1,\ldots,x_m)$ of $\mathbb{F}\lambdangle X\mathsf{R}A$ such that $f(a_1,\ldots,a_m)=0$ in $\mathcal{A}$ for all $a_1,\ldots,a_m\in \mathcal{A}$. The set $\Id{\mathcal{A}}$ of all polynomial identities for $\mathcal{A}$ is a T-ideal, i.e., $\Id{\mathcal{A}}$ is an ideal of $\FF\lambdangle X\rangle$ such that $\phi(\Id{\mathcal{A}})\subset \Id{\mathcal{A}}$ for every endomorphism $\phi$ of $\FF\lambdangle X\rangle$. An algebra that satisfies a nontrivial polynomial identity is called a PI-algebra. A T-ideal $I$ of $\mathbb{F}\lambdangle X\mathsf{R}A$ generated by $f_1,\ldots,f_k\in \mathbb{F}\lambdangle X\mathsf{R}A$ is the minimal T-ideal of $\mathbb{F}\lambdangle X\mathsf{R}A$ that contains $f_1,\ldots,f_k$. We say that $f\in \mathbb{F}\lambdangle X\mathsf{R}A$ follows from $f_1,\ldots,f_k$ if $f\in I$. Given a monomial $w\in \mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$, we write $\deltag_{x_i}(w)$ for the number of letters $x_i$ in $w$ and $\mathop{\rm mdeg}(w)$ for the multidegree $(\deltag_{x_1}(w),\ldots,\deltag_{x_m}(w))$ of $w$. An element $f\in\FF\lambdangle X\rangle$ is called multihomogeneous if it is a linear combination of monomials of the same multidegree. We say that algebras $\mathcal{A}$, $\mathcal{B}$ are called PI-equivalent and write $\mathcal{A} \sigmam_{\rm PI} \mathcal{B}$ if $\Id{\mathcal{A}} =\Id{\mathcal{B}}$.
Denote the $n^{\rm th}$ Weyl algebra by
$$\mathsf{A}W_n=\mathbb{F}\lambdangle x_1,\ldots,x_n,y_1,\ldots,y_n\mathsf{R}A/I,$$
where the ideal $I$ is generated by $[y_i,x_j]=\delta_{ij}$, $[x_i,x_j]=0$, $[y_i,y_j]=0$ for all $1\leq i,j\leq n$. Note that $\mathsf{A}_1=\mathsf{A}W_1$.
Assume that $p=0$. It is well-known that the algebra $\mathsf{A}W_n$ does not have nontrivial polynomial identities. Nevertheless, some subspaces of $\mathsf{A}W_n$ satisfy certain polynomial identities. Namely, denote by $\mathsf{A}W_n^{(1,1)}$ the $\mathbb{F}$-span of $x_i y_j$ in $\mathsf{A}W_n$ for all $1\leq i,j \leq n$ and by $\mathsf{A}W_n^{(-,r)}$ the $\mathbb{F}$-span of $a y_{j_1}\cdots y_{j_r}$ in $\mathsf{A}W_n$ for all $1\leq j_1,\ldots, j_r\leq n$ and $a\in \mathbb{F}[x_1,\ldots,x_n]$. Dzhumadil'daev~\cite{Askar_2004, Askar_2014} studied the standard polynomial
$${\rm St}_N(t_{1},\ldots,t_{N})=\sum_{\sigmagma\in S_{N}}(-1)^{\sigmagma}t_{\sigmagma(1)}\cdots t_{\sigmagma(N)}$$
over some subspaces of $\mathsf{A}W_n$. Namely, he showed that
\betagin{enumerate}
\item[$\bullet$] ${\rm St}_N$ is a polynomial identity for $\mathsf{A}W_n^{(-,1)}$ in case $N\geq n^2 + 2n$;
\item[$\bullet$] ${\rm St}_N$ is not a polynomial identity for $\mathsf{A}W_n^{(-,1)}$ in case $N< n^2 + 2n - 1$;
\item[$\bullet$] ${\rm St}_N$ is a polynomial identity for $\mathsf{A}W_1^{(-,r)}$ if and only if $N>2r$;
\item[$\bullet$] the minimal degree of nontrivial identity in $\mathsf{A}W_1^{(-,r)}$ is $2r+1$.
\end{enumerate}
Using graph--theoretic combinatorial approach Dzhumadil'daev and Yeliussizov~\cite{Askar_Yeliussizov_2015} established that
\betagin{enumerate}
\item[$\bullet$] ${\rm St}_{2n}$ is a polynomial identity for $\mathsf{A}W_n^{(1,1)}$ if and only if $n=1,2,3$.
\end{enumerate}
Note that the space $\mathsf{A}W_n^{(-,1)}$ together the multiplication given by the Lie bracket is the $n^{\rm th}$ Witt algebra $W_n$, which is a simple infinitely dimensional Lie algebra. The polynomial identities for the Lie algebra $W_n$ were studied by Mishchenko~\cite{Mishchenko_1989}, Razmyslov~\cite{Razmyslov_book} and others. The well-known open conjecture claims that all polynomial identities for $W_1$ follow from the standard Lie identity
$$\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}[[[[t_0,t_{\sigma(1)}]t_{\sigma(2)}]t_{\sigma(3)}]t_{\sigma(4)}].$$
\noindent{}$\mathbb{Z}$-graded identities for $W_1$ were described by Freitas, Koshlukov and Krasilnikov~\cite{W1_2015}. | 4,000 | 28,609 | en |
train | 0.33.2 | \subsection{Results}
In Theorem~\ref{Teorema Principal} we prove that over an infinite field $\mathbb{F}$ of positive characteristic $p$ the algebra $\mathsf{A}_h(\mathsf{B})$ is PI-equivalent to the algebra of $p\times p$ matrices over $\mathsf{B}$ in case $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$. On the other hand, over a finite field the similar result does not hold in case $\mathsf{B}=\mathbb{F}$ (see Theorem~\ref{theo_finite}). As about the case of zero characteristic, in Theorem~\ref{theo0} we prove that similarly to $\mathsf{A}_1$, the algebra $\mathsf{A}_h(\mathsf{B})$ does not have nontrivial polynomial identities.
In Section~\ref{section_example} we consider the algebra $\mathsf{A}_{h}(\mathsf{B})=\mathsf{A}_{\zeta}(\mathbb{F}^2)$ such that $h=\zeta$ does not satisfy Convention~\ref{conv1} and the statements of Theorems \ref{theo0} and ~\ref{Teorema Principal} do not hold for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. We describe polynomial identities for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$ and compare them with the polynomial identities for the Grassmann unital algebra of finite rank. | 324 | 28,609 | en |
train | 0.33.3 | \section{Properties of $\mathsf{A}_h(\mathsf{B})$}
Many properties of an Ore extension $A=\mathsf{R}[y,\sigmagma,\deltalta]$ are inherited from an underlying algebra $\mathsf{R}$. Namely, it is well-known that when $\sigmagma$ is an automorphism, then:
\betagin{itemize}
\item $A$ is a free right and a free left $\mathsf{R}$-module with the basis $\{y^i \ | \ i\geq 0\}$ (see Proposition 2.3 of~\cite{K. R. Goodearl});
\item in case $R$ is left (right, respectively) noetherian we have that $A$ is left (right, respectively) noetherian (see Theorem 2.6 of~\cite{K. R. Goodearl});
\item in case $\mathsf{R}$ is a domain we have that $A$ is a domain (see Exercise 2O~of \cite{K. R. Goodearl}).
\end{itemize}
\noindent{}In case $\mathsf{B}=\mathbb{F}$ the algebra $\mathsf{A}_h=\mathsf{A}_h(\mathsf{B})$ is a noetherian domain, but in general case $\mathsf{A}_h(\mathsf{B})$ lacks these properties, since $\mathsf{B}\subset \mathsf{A}_h(\mathsf{B})$ (see also Example~\ref{ex2} below).
In order to distinguish the generators for the algebras $\mathsf{A}_h(\mathsf{B})$ and $\mathsf{A}_1(\mathsf{B})$, we will use the following
\betagin{conv}\lambdabel{conv2} The generators of $\mathsf{A}_h(\mathsf{B})$ are denoted by $x,\widehat{y}, 1$ and the generators of $\mathsf{A}_1(\mathsf{B})$ are denoted by $x,y,1$.
\end{conv}
\betagin{lemma}\lambdabel{lemma_basis}
The sets $\{x^{i}\widehat{y}^{j} \ | \ i,j\geq 0\}$ and $\{\widehat{y}^{j}x^{i} \ | \ i,j\geq 0\}$ are $\mathsf{B}$-bases for $\mathsf{A}_h(B)$.
\end{lemma}
\betagin{proof}
Obviously, $\mathsf{A}_h(\mathsf{B})$ is the $\mathsf{B}$-span of each of the sets from the lemma. On the other hand, $\mathsf{B}$-linear independence of these sets follows from the fact that $\mathsf{A}_h(\mathsf{B})$ is a free right and a free left $\mathsf{B}[x]$-module with the basis $\{\widehat{y}^i \ | \ i\geq 0\}$.
\end{proof}
Introduce the following lexicographical order on $\mathbb{Z}^2$:
$(i,j)<(r,s)$ in case $j<s$ or $j=s,\;i<r$. Denote the multidegree of a monomial $w=x^i\widehat{y}^j$ of $\mathsf{A}_h(\mathsf{B})$ by $\mathop{\rm mdeg}(w)=(i,j)$. Given an arbitrary element
$a=\sum_{i,j\geq0}\beta_{ij} x^i \widehat{y}^j$ of $\mathsf{A}_h(\mathsf{B})$, where only finitely many $\beta_{ij}\in\mathsf{B}$ are non-zero, define its {\it multidegree} $\mathop{\rm mdeg}(a)=(d_x,d_y)$ as the maximal multidegree of its momomials, i.e. as the maximal element of the set $\{(i,j) \ | \ \beta_{ij}\neq0\}$. By Lemma~\ref{lemma_basis} the multidegree is well-defined. As above, the coefficient $\beta_{d_x,d_y}$ is called the {\it leading coefficient} of $a$ and the product $\beta_{d_x,d_y}\,x^{d_x} \widehat{y}^{d_y}$ is called the {\it leading term} of $a$. In case $a\in\mathsf{B}$ we set $\mathop{\rm mdeg}(a)=(0,0)$ and the leading coefficient as well as the leading term of $a$ is $a$.
\betagin{lemma}\lambdabel{lemma_mult}
Assume $i,j,r,s\geq 0$. Then
\betagin{enumerate}
\item[(a)] the leading term of $x^i \widehat{y}^j\cdot x^r \widehat{y}^s$ is $x^{i+r}\, \widehat{y}^{j+s}$;
\item[(b)] in case $h\in\mathsf{B}$ we have
$$x^i \widehat{y}^j\cdot x^r \widehat{y}^s = \sum_{k=0}^{\min\{j,r\}} k! \binom{j}{k} \binom{r}{k} x^{i+r-k} h^k\, \widehat{y}^{j+s-k}.$$
\end{enumerate}
\end{lemma}
\betagin{proof}
Recall that $\delta(a)=a'h$ for each $a\in\mathsf{B}[x]$. Since $[\widehat{y},a]=\delta(a)$, the induction on $j$ implies that
\betagin{eq}\lambdabel{eq1_lemma_mult}
\widehat{y}^j x^r = \sum_{k=0}^{j} \binom{j}{k} \delta^k(x^r) \widehat{y}^{j-k}
\end{eq}
(cf.~Lemma 5.2 of~\cite{Benkart_Lopes_Ondrus_I}). Taking $k=0$ in equality~\mathsf{R}ef{eq1_lemma_mult}, we obtain that the leading term of $\widehat{y}^j x^r$ is $x^r\widehat{y}^j$. Similarly we conclude the proof of part (a). Part (b) follows from~\mathsf{R}ef{eq1_lemma_mult} and
$$\delta^k(x^r) =
\left\{
\betagin{array}{cl}
\frac{r!}{(r-k)!} x^{r-k} h^k, & \text{if } k\leq r \\
0, & \text{if } k> r \\
\end{array}
\right..
$$
\end{proof}
\betagin{lemma}\lambdabel{lemma_deg}
If the leading coefficient of one of non-zero elements $a,b\in \mathsf{A}_h(\mathsf{B})$ is not a zero divisor, then $\mathop{\rm mdeg}(ab) = \mathop{\rm mdeg}(a) + \mathop{\rm mdeg}(b)$. In particular, $ab$ is not zero.
\end{lemma}
\betagin{proof} Consider $a=\sum_{i=1}^m \beta_{i} x^{r_i} \widehat{y}^{s_i}$ and $b=\sum_{j=1}^n \gamma_{j} x^{k_j} \widehat{y}^{l_j}$ for $m,n\geq1$ and non-zero $\beta_i,\gamma_j\in\mathsf{B}$, where we assume that elements of each of the sets $\{(r_i,s_i) \ | \ 1\leq i\leq m\}$ and $\{(k_j,l_j) \ | \ 1\leq j\leq n\}$ are pairwise different. Assume that $\mathop{\rm mdeg}(a)=(r_1,s_1)$ and $\mathop{\rm mdeg}(b)=(k_1,l_1)$. Part (a) of Lemma~\ref{lemma_mult} implies that
$$\mathop{\rm mdeg}(x^{r_1} \widehat{y}^{s_1} x^{k_1} \widehat{y}^{l_1})=(r_1\!+\!k_1, s_1\!+\!l_1) \;\text{ and }\; \mathop{\rm mdeg}(x^{r_i} \widehat{y}^{s_i} x^{k_j} \widehat{y}^{l_j})<(r_1\!+\!k_1, s_1\!+\!l_1)$$ if $(i,j)\neq (1,1)$. Since $\beta_1 \gamma_1\neq 0$, we obtain $\mathop{\rm mdeg}(ab) = (r_1+k_1, s_1 + l_1)$ and the proof is concluded.
\end{proof}
\betagin{lemma}\lambdabel{lemma_embedding}
\betagin{enumerate}
\item[(a)] The $\mathsf{B}$-linear homomorphism of $\mathbb{F}$-algebras $\psi: \mathsf{A}_h(\mathsf{B})\to \mathsf{A}_1(\mathsf{B})$, defined by
$$1\to 1,\quad x\to x,\quad \widehat{y}\to y h,$$
is an embedding $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$.
\item[(b)] $\{x^{i}h^{j}y^{j} \ | \ i,j\geq 0\}$ and $\{y^{j}h^{j}x^{i} \ | \ i,j\geq 0\}$ are $\mathsf{B}$-bases for $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$.
\end{enumerate}
\end{lemma}
\betagin{proof}
\noindent{\bf (a)} Since $\psi([\widehat{y},x]-h)=([y,x]-1)h=0$ in $\mathsf{A}_1(\mathsf{B})$, the homomorphism $\psi$ is well-defined. Assume that $\psi$ is not an embedding, i.e., there exists non-zero finite sum $a=\sum_{i,j\geq0}\beta_{ij} x^i \widehat{y}^j$ with $\beta_{ij}\in \mathsf{B}$ such that
$$
\psi(a)=\sum_{i,j\geq0}\beta_{ij} x^i (y h)^j =0 \quad\text{ in }\quad\mathsf{A}_1(\mathsf{B}).
$$
Denote $\mathop{\rm mdeg}(a)=(r,s)$. If $(r,s)=(0,0)$, then $a\in \mathsf{B}$ and $\psi(a)=a$ is not zero; a contradiction. Therefore, $(r,s)\neq (0,0)$.
Since $\mathop{\rm mdeg}(x^i (y h)^j)= (i + j \deltag(h), j)$ by Lemma~\ref{lemma_deg} and Convention~\ref{conv1}, we obtain that $\mathop{\rm mdeg}(\psi(a))= (r + s \deltag(h), s)$ is not zero; a contradiction.
\noindent{\bf (b)} Since $h$ lies in the center of $\mathsf{B}[x]$, repeating the proof of Lemma 3.4 from~\cite{Benkart_Lopes_Ondrus_I} for $\mathsf{A}_h(\mathsf{B})$ we can see that
$$\mathsf{A}_h(\mathsf{B})=\bigoplus_{j\geq 0}\mathsf{B}[x]h^{j}y^{j}=\bigoplus_{j\geq 0}y^{j}h^{j}\mathsf{B}[x].$$
Similarly to part (a), we conclude the proof by the reasoning with multidegree.
\end{proof}
\betagin{example}\lambdabel{ex2} Assume $\mathsf{B}$ is the $\mathbb{F}$-algebra of double numbers, i.e., $\mathsf{B}$ has an $\mathbb{F}$-basis $\{1,\zeta\}$ with $\zeta^2=0$. Then the ideal $I=\mathbb{F}\text{-span}\{\zeta x^i \widehat{y}^j \ | \ i,j\geq 0\}$ is a proper nilpotent ideal of $\mathsf{A}_h(\mathsf{B})$. In particular, the algebra $\mathsf{A}_h(\mathsf{B})$ is not semi-prime.
\end{example} | 2,992 | 28,609 | en |
train | 0.33.4 | \section{$\mathsf{A}_h(\mathsf{B})$ as the algebra of differential operators}\lambdabel{section_diff}
Denote by ${\rm Map}(\mathsf{B}[z])$ the algebra of all $\mathbb{F}$-linear maps over $\mathsf{B}[z]$ with respect to composition. Assume that $\mathfrak{D}_h(\mathsf{B}[z])$ is the subalgebra of ${\rm Map}(\mathsf{B}[z])$ generated by the following maps: the multiplication $\beta\,{\rm Id}$ by an element $\beta$ of $\mathsf{B}$, i.e., $(\beta\,{\rm Id})(f)=\beta f$, the multiplication $\chi$ by $z$, i.e., $\chi(f)=zf$, and the derivation $\delta$ given by $\delta(f)=f'h(z)$ for all $f\in\mathsf{B}[z]$. Note that $\delta= h(\chi)\, \partial$, where $\partial$ stands for the operator of the usual derivative. Obviously, maps $\chi$, $h(\chi)$ and $\partial$ are $\mathsf{B}$-linear. For short, we write $\chi^0$ for ${\rm Id}$.
\betagin{prop}\lambdabel{prop_diff}
\betagin{enumerate}
\item[(a)] $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ i,j\geq 0\}$ is an $\mathsf{B}$-basis for $\mathfrak{D}_h(\mathsf{B}[z])$ in case $p=0$.
\item[(b)] $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ 0\leq i,\; 0\leq j<p \}$ is an $\mathsf{B}$-basis for $\mathfrak{D}_h(\mathsf{B}[z])$ in case $p>0$.
\item[(c)]$\mathsf{A}_h(\mathsf{B}) /\id{h^p y^p} \sigmameq \mathfrak{D}_h(\mathsf{B}[z])$ for each $p\geq 0$.
\end{enumerate}
\end{prop}
\betagin{proof}
Consider the $\mathsf{B}$-linear homomorphism of $\mathbb{F}$-algebras $\Phi: \mathsf{A}_1(\mathsf{B}) \to {\rm Map}(\mathsf{B}[z])$ given by
$1\to {\rm Id}$, $x\to \chi$, $y\to \partial$. Since $\Phi([y,x]-1)(f)=(\partial \chi - \chi \partial - {\rm Id})(f) = f + z f' - z f' - f = 0$ for all $f\in\mathsf{B}[z]$, the map $\Phi$ is a well-defined. Applying $\Phi$ to parts (a) and (b) of Lemma~\ref{lemma_embedding} we obtain that $\mathfrak{D}_h(\mathsf{B}[z]) = \Phi(\mathsf{A}_h(\mathsf{B}))$ is an $\mathsf{B}$-span of $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ i,j\geq 0\}$.
Let $p=0$. Assume that some non-zero finite sum $\pi=\sum_{i,j\geq 0} \beta_{ij} \chi^{i} h(\chi)^j \partial^{j}$ with $\beta_{ij}\in\mathsf{B}$ belongs to the kernel of $\Phi$. Denote by $j_0$ the minimal $j\geq0$ with $\beta_{ij}\neq 0$ for some $i$ and denote by $i_0$ the maximal $i$ with $\beta_{ij_0}\neq 0$. Then $\pi(z^{j_0}) = j_0!\, h(z)^{j_0} \sum_{0\leq i\leq i_0} \beta_{i,j_0}\, z^i =0$ in $\mathsf{B}[z]$. Thus Convention~\ref{conv1} together with $j_0!\,\beta_{i_0,j_0} \neq 0$ implies a contradiction. Part~(a) is proven.
Assume that $p>0$ and some non-trivial finite sum
$$\pi=\sum\limits_{0\leq i,\; 0\leq j<p } \beta_{ij} \chi^{i} h(\chi)^j \partial^{j}$$
with $\beta_{ij}\in\mathsf{B}$ belongs to the kernel of $\Phi$. As above, we obtain a contradiction. Since $\partial^p=0$, we conclude the proof of part (b).
Parts (a) and (b) together with part (b) of Lemma~\ref{lemma_embedding} conclude the proof of part (c).
\end{proof}
\betagin{theo}\lambdabel{theo0}
In case $p=0$ the algebra $\mathsf{A}_h(\mathsf{B})$ does not have nontrivial polynomial identities.
\end{theo}
\betagin{proof}
Assume that $\mathbb{F}$-algebra $\mathsf{A}_h(\mathsf{B})$ has a nontrivial polynomial identity. Since $p=0$, there exists $N>0$ such that $\mathsf{A}_h(\mathsf{B})$ satisfies a nontrivial multilinear identity $f(x_1,\ldots,x_N)=\sum_{\sigma\in S_N}\alpha_{\sigma} x_{\sigma(1)}\cdots x_{\sigma(N)}$ with $\alpha_{\sigma}\in \mathbb{F}$. Moreover, we can assume that $\alpha_{\rm Id}\neq0$ for the identity permutation ${\rm Id}$. Given $j>0$, we write $F_j$ for a $\mathsf{B}$-linear map $\chi^{2j} h(\chi)^j \partial^{j}$ from $\mathfrak{D}_h(\mathsf{B}[z])$. Denote by $d\geq0$ the degree of $h$ and we write $\eta$ for the leading coefficient of $h$. Recall that $\eta$ is not a zero divisor by Convention~\ref{conv1}. Note that
\betagin{eq}\lambdabel{eq_theo0}
F_{j}(z^m)=\left\{
\betagin{array}{cc}
0, & \text{ in case }m<j\\
\frac{m!}{(m-j)!} \, z^{m+j} h(z)^j, & \text{ in case }m\geq j.\\
\end{array}
\right.
\end{eq}
In particular, $\deltag(F_{j}(z^j))=j(d+2)$ and the leading coefficient of $F_{j}(z^j)$ is $j!\,\eta^j$, which is not a zero divisor.
By parts (a) and (c) of Proposition~\ref{prop_diff}, the equality $f(F_{j_{1}},\ldots,F_{j_N})=0$ holds in $\mathfrak{D}_h(\mathsf{B}[z])$ for all $j_1,\ldots,j_{N}>0$. Consider $j_k=(d+2)^{N-k}$ for all $0\leq k\leq N$.
Note that $1=j_N<j_{N-1}<\cdots < j_0$. We claim that for any $\sigma\in S_N$ we have
\betagin{eq}\lambdabel{claim_theo0}
F_{j_{\sigma(1)}}\circ \cdots \circ F_{j_{\sigma(N)}}(z) \neq 0 \text{ if and only if } \sigma= {\rm Id}.
\end{eq}
\betagin{eq}\lambdabel{claim2_theo0}
\text{The leading term of } F_{j_1}\circ \cdots \circ F_{j_N}(z) \text{ is } j_{1}!\cdots j_N!\,\eta^{j_{1}+\cdots +j_N} z^{j_{0}}.
\end{eq}
Let us prove these claims. Assume that $F_{j_{\sigma(1)}}\circ \cdots \circ F_{j_{\sigma(N)}}(z) \neq 0$ for some $\sigma\in S_N$.
Since $F_{j_{\sigma(N)}}(z)\neq 0$, then equality~\mathsf{R}ef{eq_theo0} implies that $\sigma(N)=N$, $j_{\sigma(N)}=1$, $\deltag(F_{j_{\sigma(N)}}(z))=d+2=j_{N-1}$ and the leading coefficient of $F_{j_{\sigma(N)}}(z)$ is $\eta$, which is not a zero divisor.
Similarly, assume that for $1\leq l< N$ with $\sigma(l)\leq l$ the inequality $F_{j_{\sigma(l)}}(g)\neq 0$ holds for some $g\in \mathsf{B}[z]$ with the leading term $j_{l+1}!\cdots j_N!\,\eta^{j_{l+1}+\cdots + j_N} z^{j_l}$. Then equality~\mathsf{R}ef{eq_theo0} implies that $\sigma(l)=l$ and $\deltag(F_{j_{\sigma(l)}}(g))=j_{l-1}$. Moreover, the leading term of $F_{j_{\sigma(l)}}(g)$ is $j_{l}!\cdots j_N!\,\eta^{j_{l}+\cdots +j_N} z^{j_{l-1}}$. Consequently applying this reasoning to $l=N-1,\, l=N-2,\ldots, l=1$, we conclude the proof of claims~\mathsf{R}ef{claim_theo0} and~\mathsf{R}ef{claim2_theo0}.
Claims~\mathsf{R}ef{claim_theo0} and~\mathsf{R}ef{claim2_theo0} imply that $0=f(F_1,\ldots,F_N)(z)=\alpha_{\rm Id}\, F_{j_1}\circ \cdots \circ F_{j_N}(z)\neq 0$ by Convention~\ref{conv1}; a contradiction.
\end{proof} | 2,437 | 28,609 | en |
train | 0.33.5 | \section{Polynomial Identities for $\mathsf{A}_h(\mathsf{B})$ in positive characteristic}\lambdabel{section_positive}
We write $M_n=M_n(\mathbb{F})$ for the algebra of all $n\times n$ matrices over $\mathbb{F}$ and denote by $\widetilde{M}_n$ the algebra of all $n\times n$ matrices over $\mathsf{B}[x,y]$. Denote by $I_n$ the identity $n\times n$ matrix and by $E_{ij}\in M_n$ the matrix such that the $(i,j)^{\rm th}$ entry is equal to one and the rest of entries are zeros. Consider the properties of the next two matrices of $M_p$:
$$A_{0}=\sum_{i=1}^{p-1}E_{i+1,i}\;\;\text{ and }\;\;B_{0}=\sum_{i=1}^{p-1}i\cdot E_{i,i+1}.$$
\betagin{lemma}\lambdabel{lemma_A0_B0}
\betagin{enumerate}
\item[(a)] For all $0\leq k<p$ we have that
$$A_{0}^{k}=\sum_{i=1}^{p-k}E_{k+i,i}\;\;\text{ and }\;\; B_{0}^{k}=\sum_{i=1}^{p-k}\frac{(k+i-1)!}{(i-1)!}E_{i,k+i},$$
\noindent{}where $A_0^0$ and $B_0^0$ stand for $I_p$.
\item[(b)] $B_{0}A_{0}-A_{0}B_{0}=I_{p}$.
\end{enumerate}
\end{lemma}
\betagin{proof} The formula for $A_{0}^{k}$ is trivial. We prove the formula for $B_{0}^{k}$ by induction on $k$. For $k=1$ the claim holds. Assume that the claim is valid for some $k<p-1$. Then
\betagin{align*}
B_{0}^{k+1}&=\left(\sum_{i=1}^{p-k}\frac{(k+i-1)!}{(i-1)!}E_{i,k+i}\right)\left(\sum_{r=1}^{p-1}r\cdot E_{r,r+1} \right)=\sum_{i=1}^{p-(k+1)}\frac{(i+k)!}{(i-1)!}E_{i,k+1+i}
\end{align*}
and the required is proven. Part (b) is straightforward.
\end{proof}
Define the $\mathsf{B}$-linear homomorphism $\varphi:\mathsf{B}\lambdangle x,y \mathsf{R}A \rightarrow \widetilde{M}_p$ of algebras by
$$x\mapsto A,\;\; y\mapsto B,\;\; 1\mapsto I_p,$$
where $A=xI_{p}+A_{0}$ and $B=yI_{p}+B_{0}$. Since $\beta A=A \beta$ and $\beta B = B\beta$ for each $\beta\in B$, the homomorphism $\varphi$ is well-defined.
\betagin{lemma}\lambdabel{A1 subset Map}
The homomorphism $\varphi$ induces the injective $\mathsf{B}$-linear homomorphism $\ov{\varphi}:\mathsf{A}_1(\mathsf{B})\rightarrow \widetilde{M}_p$. In particular, the restriction of $\ov{\varphi}$ to $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$ is the injective $\mathsf{B}$-linear homomorphism $\mathsf{A}_h(\mathsf{B})\rightarrow \widetilde{M}_p$.
\end{lemma}
\betagin{proof}
By part (b) of Lemma \ref{lemma_A0_B0} we have that $\varphi(yx-xy-1)=BA-AB-I_{p}=0$. Therefore, $\varphi$ induces a $\mathsf{B}$-linear homomorphism $\ov{\varphi}:\mathsf{A}_1(\mathsf{B})\rightarrow \widetilde{M}_p$ of algebras.
Assume that there exists a nonzero $a\in \mathsf{A}_1(\mathsf{B})$ such that $\ov{\varphi}(a)=0$. Since $\{x^{i}y^{j} \ | \ i,j\geq 0\}$ is an $\mathsf{B}$-basis for $\mathsf{A}_1(\mathsf{B})$ by Lemma~\ref{lemma_basis}, we have $a=\sum_{i,j\geq 0}\beta_{ij}x^{i}y^{j}$ for a finite sum with $\beta_{ij}\in \mathsf{B}$. Thus
$0= \ov{\varphi}(a) = \sum_{i,j\geq 0} \beta_{ij} A^i B^j$. The equalities
$$A^{i}=\betagin{pmatrix}
x^{i} & 0 &\cdots & 0 \\
* & * &\cdots & * \\
\vdots &\vdots&&\vdots\\
* & * &\cdots & * \\
\end{pmatrix}
\;\; \text{and} \;\;
B^{j}=\betagin{pmatrix}
y^{j} & * &\cdots & * \\
0 & * &\cdots & * \\
\vdots &\vdots&&\vdots\\
0 & * &\cdots & * \\
\end{pmatrix}$$
imply that the $(1,1)^{\rm th}$ entry of $A^{i}B^{j}$ is $(A^{i}B^{j})_{1,1}=x^{i}y^{j}$. Hence
$0 = (\varphi(a))_{1,1} = \sum_{i,j\geq 0} \beta_{ij} x^{i}y^{j}$ in $\mathsf{B}[x,y]$.
Hence $\beta_{ij}=0$ for all $i,j\geq 0$ and $a=0$; a contradiction. Therefore $\ov{\varphi}$ is injective.
\end{proof}
Given $1\leq i,j\leq p$ and $k\geq 1$, we write $z_{ij}(k)$ for $x_{i+p(j-1)+p^2(k-1)}\in \mathbb{F}\lambdangle X\mathsf{R}A$. The {\it generic $p\times p$ matrix $X_k$ with non-commutative elements} is the matrix $X_k=(z_{ij}(k))_{1\leq i,j\leq p}$.
\betagin{cor}\lambdabel{Id(Mp) subset Id(Ah)}
$\Id{M_p(\mathsf{B})}\subset \Id{\mathsf{A}_h(\mathsf{B})}$, if the field $\mathbb{F}$ is infinite.
\end{cor}
\betagin{proof}
Lemma~\ref{A1 subset Map} implies that $\Id{\widetilde{M}_p}\subset \Id{\mathsf{A}_h(\mathsf{B})}$.
Since $B\subset B[x,y]$, we have $\Id{\widetilde{M}_p}\subset \Id{M_p(\mathsf{B})}$. On the other hand, assume that $f=f(x_1,\ldots,x_n)$ is a polynomial identity for $M_p(\mathsf{B})$. Then $f(X_1,\ldots,X_n)= (f_{ij})_{1\leq i,j\leq n}$ for some $f_{ij}\in \mathbb{F}\lambdangle X\mathsf{R}A$ with $f_{ij}\in \Id{\mathsf{B}}$. It is well-known that for an infinite field $\mathbb{F}$ and a commutative unital $\mathbb{F}$-algebra $\mathcal{C}$ the polynomial identities for a unital $\mathbb{F}$-algebra $\mathcal{B}$ and $\mathcal{C}\otimes_{\mathbb{F}} \mathcal{B}$ are the same (for example, see Lemma 1.4.2 of~\cite{A.Gia}). Since $\mathsf{B}[x,y]=\mathbb{F}[x,y]\otimes_{\mathbb{F}} \mathsf{B}$, we obtain $f_{ij}\in \Id{\mathsf{B}[x,y]}$ and $f\in \Id{\widetilde{M}_p}$. The required is proven.
\end{proof}
For each $\alpha\in Z(\mathsf{B})$ consider the evaluation $\mathsf{B}$-linear homomorphism $\epsilon_{\alpha}:\mathsf{B}[x,y]\rightarrow \mathsf{B}$ of unital $\mathbb{F}$-algebras defined by
$$x\mapsto \alpha, \;\; \ y\mapsto 0$$
and extend it to the evaluation homomorphism $\varepsilon_{\alpha}:\widetilde{M}_p\rightarrow M_p(\mathsf{B})$. Since $\mathsf{A}_h(\mathsf{B})$ is a subalgebra of $\widetilde{M}_p$ by means of embedding $\ov{\varphi}$ (see Lemma \ref{A1 subset Map}), we can consider the images of $x,\widehat{y}\in \mathsf{A}_h(\mathsf{B})$ in $M_p(\mathsf{B})$, which we denote by $C_{\alpha}$ and $D_{\alpha}$, respectively:
$$\betagin{array}{cll}
C_{\alpha}=\varepsilon_{\alpha}(\ov{\varphi}(x))& =\varepsilon_{\alpha}(A) & = \alpha I_p + A_0,\\
D_{\alpha}=\varepsilon_{\alpha}(\ov{\varphi}(\widehat{y}))& =\varepsilon_{\alpha}(Bh(A)) & = B_{0}\varepsilon_{\alpha}(h(A)).
\end{array}
$$
\noindent{}Obviously, $\beta C_{\alpha} = C_{\alpha} \beta$ and $\beta D_{\alpha} = D_{\alpha} \beta$ for each $\beta\in B$. To obtain the explicit description of the matrix $D_{\alpha}$ we calculate $h(A)$. For $r\geq1$ denote the $r^{\rm th}$ derivative of $h\in\mathsf{B}[x]$ by $h^{(r)}=\frac{d^{r}h}{dx^{r}}$ and write $h^{(0)}$ for $h$. Note that $h^{(r)}(\alpha)$ lies in the center of $\mathsf{B}$.
\betagin{lemma}\lambdabel{lemma_hA}
$$h(A) =\sum_{i=1}^{p}\sum_{j=1}^{i} \frac{1}{(i-j)!} h^{(i-j)} E_{ij}.$$
\end{lemma}
\betagin{proof}
We start with the case of $h=x^{k}\in\mathsf{B}[x]$ for some $k\geq0$. Obviously, the claim of the lemma holds for $h=1$. Therefore, we assume that $k\geq1$. Since $A_{0}^{r}=0$ for all $r\geq p$, we have
$$h(A) = A^{k} = (xI_{p}+A_{0})^{k} = \sum_{r=0}^{\min\{k,p-1\}}\binom{k}{r}x^{k-r}A_{0}^{r}.$$
\noindent{}Part (a) of Lemma \ref{lemma_A0_B0} implies
$$h(A) = \sum_{r=0}^{\min\{k,p-1\}}\binom{k}{r}x^{k-r} \left(\sum_{i=1}^{p-r}E_{r+i,i}\right).$$
\noindent{}Regrouping the terms we obtain
\betagin{eq}\lambdabel{eq_lemma_hA}
h(A)=\sum_{i=1}^{p}\; \sum_{j=\max\{1,i-k\}}^{i} \binom{k}{i-j}x^{k-(i-j)}E_{ij}.
\end{eq}
\noindent{}Note that for $0\leq r< p$ we can rewrite
$$\frac{1}{r!} h^{(r)} = \left\{
\betagin{array}{rl}
\binom{k}{r}x^{k-r}, & \text{ if } r\leq k \\
0, & \text{ otherwise } \\
\end{array}
\right.,
$$
where $r!$ is not zero in $\mathbb{F}$. Hence equality~\mathsf{R}ef{eq_lemma_hA} implies that the claim holds for $h=x^{k}$.
The general case follows from the proven partial case and the $\mathsf{B}$-linearity of derivatives.
\end{proof}
Lemma~\ref{lemma_hA} together with the definition of $D_{\alpha}$ immediately implies the next corollary.
\betagin{cor}\lambdabel{cor_D0}
$$D_{\alpha} = \sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \frac{i}{(i-j+1)!} h^{(i-j+1)}(\alpha) E_{ij}.$$
\end{cor}
For short, denote the $(i,j)^{\rm th}$ entry of $D_{\alpha}$ by $\D{i}{j}\in Z(\mathsf{B})$ and for all $1\leq k < p$ define
\betagin{eq}\lambdabel{eq_Dak}
D_{\alpha,k} = D_{\alpha} - \sum_{r=0}^{k-1}\D{k}{k-r}A_0^{r} = D_{\alpha} - \sum_{r=0}^{k-1} \sum_{i=1}^{p-r} \D{k}{k-r}E_{r+i,i}.
\end{eq}
\noindent{}We apply the following technical lemma in the proof of key Proposition~\ref{Id(Ah) subset Id(Mp)} (see below).
\betagin{lemma}\lambdabel{Epk.Dk}
For all $1\leq r\leq p$ and $1\leq k < p$ we have
$$E_{rk}\, D_{\alpha,k}=k\, h(\alpha) E_{r,k+1}.$$
\end{lemma}
\betagin{proof}
We have
\betagin{align*}
E_{rk}D_{\alpha,k}&=\sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \D{i}{j}E_{rk}E_{ij} - \sum_{j=0}^{k-1}\sum_{i=1}^{p-j} \D{k}{k-j} E_{rk} E_{i+j,i}\\
&= \sum_{j=1}^{k+1} \D{k}{j}E_{rj} - \sum_{j=1}^{k} \D{k}{j} E_{rj}\\
&= \D{k}{k+1}E_{r,k+1}.\\
\end{align*}
Equality $\D{k}{k+1} = k\, h(\alpha)$ concludes the proof.
\end{proof} | 3,679 | 28,609 | en |
train | 0.33.6 | Lemma~\ref{lemma_hA} together with the definition of $D_{\alpha}$ immediately implies the next corollary.
\betagin{cor}\lambdabel{cor_D0}
$$D_{\alpha} = \sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \frac{i}{(i-j+1)!} h^{(i-j+1)}(\alpha) E_{ij}.$$
\end{cor}
For short, denote the $(i,j)^{\rm th}$ entry of $D_{\alpha}$ by $\D{i}{j}\in Z(\mathsf{B})$ and for all $1\leq k < p$ define
\betagin{eq}\lambdabel{eq_Dak}
D_{\alpha,k} = D_{\alpha} - \sum_{r=0}^{k-1}\D{k}{k-r}A_0^{r} = D_{\alpha} - \sum_{r=0}^{k-1} \sum_{i=1}^{p-r} \D{k}{k-r}E_{r+i,i}.
\end{eq}
\noindent{}We apply the following technical lemma in the proof of key Proposition~\ref{Id(Ah) subset Id(Mp)} (see below).
\betagin{lemma}\lambdabel{Epk.Dk}
For all $1\leq r\leq p$ and $1\leq k < p$ we have
$$E_{rk}\, D_{\alpha,k}=k\, h(\alpha) E_{r,k+1}.$$
\end{lemma}
\betagin{proof}
We have
\betagin{align*}
E_{rk}D_{\alpha,k}&=\sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \D{i}{j}E_{rk}E_{ij} - \sum_{j=0}^{k-1}\sum_{i=1}^{p-j} \D{k}{k-j} E_{rk} E_{i+j,i}\\
&= \sum_{j=1}^{k+1} \D{k}{j}E_{rj} - \sum_{j=1}^{k} \D{k}{j} E_{rj}\\
&= \D{k}{k+1}E_{r,k+1}.\\
\end{align*}
Equality $\D{k}{k+1} = k\, h(\alpha)$ concludes the proof.
\end{proof}
\betagin{prop}\lambdabel{Id(Ah) subset Id(Mp)}
\betagin{enumerate}
\item[(a)] Assume $\alpha\in Z(\mathsf{B})$. Then $\varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B}))$ contains $h(\alpha)^{2(p-1)} M_p(\mathsf{B})$.
\item[(b)] Assume $h(\alpha)$ is invertible in $\mathsf{B}$ for some $\alpha\in Z(\mathsf{B})$. Then $\Id{\mathsf{A}_h(\mathsf{B})}\subset \Id{M_p(\mathsf{B})}$.
\item[(c)] Assume $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$ and $\mathbb{F}$ is infinite. Then $\Id{\mathsf{A}_h(\mathsf{B})}\subset \Id{M_p(\mathsf{B})}$.
\end{enumerate}
\end{prop}
\betagin{proof} For short, we write $\beta$ for $h(\alpha)\in Z(\mathsf{B})$.
\noindent{\bf (a)}
Denote by $\mathcal{L} = \varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B})) = \alphag_{\mathsf{B}}\{I_p,C_{\alpha}, D_{\alpha}\}$ the $\mathbb{F}$-algebra generated by $\mathsf{B} I_p$, $C_{\alpha}$, $D_{\alpha}$. Since $A_0=C_{\alpha} - \alpha I_p$, we obtain that
$$A_0^{k}=\sum_{i=1}^{p-k}E_{k+i,i}\in \mathcal{L}$$
for all $0\leq k<p$. In particular, $E_{p1}=A_0^{p-1}\in\mathcal{L}$. Equality~\mathsf{R}ef{eq_Dak} implies that $D_{\alpha,k}\in\mathcal{L}$ for all $1\leq k < p$.
The statement of part (a) follows from the following claim:
\betagin{eq}\lambdabel{claim1}
\{ \beta^{p+k-r-1} E_{rk} \ | \ 1\leq r,k \leq p\}\subset \mathcal{L}.
\end{eq}
\noindent{}To prove the claim we use descending induction on $r$.
Assume $r=p$. We have $E_{p1}\in\mathcal{L}$. Lemma~\ref{Epk.Dk} implies that $E_{p1} D_{\alpha,1} = \beta E_{p2}$. Since $E_{p1}$, $D_{\alpha,1}$ belong to $\mathcal{L}$, we can see that $\beta E_{p2}\in\mathcal{L}$. Similarly, the equality $\beta E_{p2}D_{\alpha,2} = 2 \beta^2 E_{p3}$ implies $\beta^2 E_{p3}\in\mathcal{L}$. Repeating this reasoning we obtain that $\beta^{k-1} E_{pk}\in\mathcal{L}$ for all $1\leq k\leq p$.
Assume that for some $1\leq r< p$ claim~\mathsf{R}ef{claim1} holds for all $r'>r$, i.e., for every $1\leq k\leq p$ we have $\beta^{p+k-r'-1} E_{r'k}\in\mathcal{L}$. Since
$$\beta^{p-r}\left(A_0^{r-1} - \sum_{k=2}^{p-r+1} E_{(r-1)+k,k}\right) = \beta^{p-r} E_{r1},$$
we obtain $\beta^{p-r} E_{r1}\in\mathcal{L}$. Lemma~\ref{Epk.Dk} implies that $\beta^{p-r} E_{r1} D_{\alpha,1} = \beta^{p+1-r} E_{r2}$. Hence $\beta^{p+1-r} E_{r2}\in\mathcal{L}$. Repeating this reasoning we obtain that $\beta^{p+k-r-1} E_{rk} \in\mathcal{L}$ for all $1< k\leq p$, since $\beta^{p+k-r-2} E_{r,k-1} D_{\alpha,k-1} = (k-1)\, \beta^{p+k-r-1} E_{r,k}$. Claim~\mathsf{R}ef{claim1} is proven.
\noindent{\bf (b)} Since $h(\alpha)$ is invertible in $\mathsf{B}$, part (a) implies that $\varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B}))=M_p(\mathsf{B})$. Since $\varepsilon_{\alpha}$ is a homomorphism of $\mathbb{F}$-algebras, the required is proven.
\noindent{\bf (c)} Consider a polynomial identity $f\in \mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ for $\mathsf{A}_h(\mathsf{B})$. Since $\mathbb{F}$ is infinite, without loss of generality we can assume that $f$ is homogeneous with respect to the natural grading of $\mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ by degrees, i.e., each monomial of $f$ has one and the same degree $t>0$.
Part (a) implies that for every $A_1,\ldots,A_m$ from $M_p(\mathsf{B})$ there exist $a_1,\ldots,a_m$ from $A_h(\mathsf{B})$ such that
$$\beta^{2(p-1)t} f(A_1,\ldots,A_m) = f(\beta^{2(p-1)} A_1,\ldots,\beta^{2(p-1)} A_m) = f(\varepsilon_{\alpha}(a_1),\ldots,\varepsilon_{\alpha}(a_m)).$$
Since $\varepsilon_{\alpha}$ is a homomorphism of $\mathbb{F}$-algebras, we have $f(\varepsilon_{\alpha}(a_1),\ldots,\varepsilon_{\alpha}(a_m))=0$. Therefore $f$ is a polynomial identity for $\mathsf{A}_h(\mathsf{B})$ because $\beta$ is not a zero divisor.
\end{proof}
To illustrate the proof of part (a) of Proposition~\ref{Id(Ah) subset Id(Mp)}, we repeat it in the partial case of $p=3$ in the following example.
\betagin{example}\lambdabel{ex1}
Assume $p = 3$ and $h(\alpha)\neq0$ for some $\alpha\in Z(\mathsf{B})$. For short, denote $\beta=h(\alpha)$, $\beta'=h'(\alpha)$ and $\beta''=h''(\alpha)$. Then $A_0=E_{21} + E_{32}$,
$$ C_{\alpha}=\betagin{pmatrix}
\alpha& 0 & 0\\
1 & \alpha & 0\\
0 & 1 & \alpha \\
\end{pmatrix},\quad\text{ and }\quad
D_{\alpha}=\betagin{pmatrix}
\beta'&\beta&0\\
\beta''& 2\beta' & 2\beta\\
0&0&0
\end{pmatrix}.
$$
To show that $\mathcal{L} = \alphag_{\mathsf{B}}\{I_p,C_{\alpha}, D_{\alpha}\}$ contains $\beta^4 M_3(\mathsf{B})$, we consider the following elements of $\mathcal{L}$:
$$ D_{\alpha,1}=D_{\alpha} - \beta' I_3=
\betagin{pmatrix}
0 & \beta & 0\\
\beta'' & \beta' & 2 \beta\\
0 & 0 & -\beta' \\
\end{pmatrix},
$$
$$ D_{\alpha,2}=D_{\alpha} - 2 \beta' I_3 - \beta'' A_0 =
\betagin{pmatrix}
-\beta' & \beta & 0\\
0 & 0 & 2 \beta\\
0 & -\beta'' & -2 \beta' \\
\end{pmatrix}.
$$
\noindent{}Note that $A_0=C_{\alpha} - \alpha I_3$ and $A_0^2=E_{31}$ belong to $\mathcal{L}$. Since
$$E_{31} D_{\alpha,1} = \beta E_{32}\;\;\text{ and }\;\; \beta E_{32} D_{\alpha,2} = 2 \beta^2 E_{33},$$
we obtain that $ \beta E_{32},\, \beta^2E_{33}\in\mathcal{L}$. Thus $ \beta(A_0-E_{32})= \beta E_{21}$ lies in $\mathcal{L}$. Since
$$\beta E_{21} D_{\alpha,1} = \beta^2 E_{22}\;\;\text{ and }\;\; \beta^2 E_{22} D_{\alpha,2} = 2 \beta^3 E_{23},$$
we obtain
that $ \beta^2 E_{22},\, \beta^3 E_{23}\in\mathcal{L}$. Hence $ \beta^2(I_3 - E_{22} - E_{33}) = \beta^2 E_{11}$ lies in $\mathcal{L}$. Since
$$\beta^2 E_{11} D_{\alpha,1} = \beta^3 E_{12}\;\;\text{ and }\;\; \beta^3 E_{12} D_{\alpha,2} = 2 \beta^4 E_{13},$$ we obtain
that $\beta^3 E_{12},\, \beta^4 E_{13}\in\mathcal{L}$. Therefore, $\mathcal{L}$ contains $\beta^4 M_3(\mathsf{B})$.
\end{example}
\betagin{theo}\lambdabel{Teorema Principal}
Assume that $\mathbb{F}$ is an infinite field of characteristic $p>0$.
\betagin{enumerate}
\item[(a)] If $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$, then $\mathsf{A}_h(\mathsf{B})\sigmam_{\rm PI} M_p(\mathsf{B})$.
\item[(b)] If $\mathsf{B}=\mathbb{F}$, then $\mathsf{A}_h\sigmam_{\rm PI} M_p$.
\end{enumerate}
\end{theo}
\betagin{proof} Part (a) follows from Corollary \ref{Id(Mp) subset Id(Ah)} and part (c) of Proposition \ref{Id(Ah) subset Id(Mp)}.
Assume that $\mathsf{B}=\mathbb{F}$. Then there exists $\alpha\in\mathbb{F}$ with $h(\alpha)\neq0$, because $h\in\mathbb{F}[x]$ is not zero and $\mathbb{F}$ is infinite. Part (a) concludes the proof of part (b).
\end{proof}
\betagin{cor}\lambdabel{cor1}
Assume that $\mathbb{F}$ is an infinite field of characteristic $p>0$ and for $h= \eta_d x^d + \eta_{d-1} x^{d-1} + \cdots + \eta_0$ from $Z(\mathsf{B})[x]$ we have that $\eta_d$ and $\eta_0$ are not zero divisors. Then $\mathsf{A}_h(\mathsf{B})\sigmam_{\rm PI} M_p(\mathsf{B})$.
\end{cor} | 3,456 | 28,609 | en |
train | 0.33.7 | \section{$\mathsf{A}_h$ over finite fields}\lambdabel{section_finite}
In this section we assume that $\mathsf{B}=\mathbb{F}$ is the field of finite of order $q=p^k$ and $\mathbb{F}\subset \mathbb{K}$ for an infinite field $\mathbb{K}$. Since $h\in\mathbb{F}[x]$, Convention~\ref{conv1} is equivalent to the inequality $h\neq0$. As above, we write $M_p$ for $M_p(\mathbb{F})$ and $\mathsf{A}_h$ for $\mathsf{A}_h(\mathbb{F})$. Given a $\mathbb{K}$-algebra $\mathcal{A}$, we write $\IdK{\mathcal{A}}$ for the ideal of $\mathbb{K}\lambdangle X\mathsf{R}A$ of polynomial identities for $\mathcal{A}$ over $\mathbb{K}$. In this section we proof the next result.
\betagin{theo}\lambdabel{theo_finite}
\betagin{enumerate}
\item[(a)] $\IdK{M_p(\mathbb{K})} \bigcap \mathbb{F}\lambdangle X\mathsf{R}A \subset \Id{\mathsf{A}_h}$.
\item[(b)] $\Id{\mathsf{A}_h} \subset \Id{M_p}$, if $h(\alpha)\neq0$ for some $\alpha\in\mathbb{F}$.
\item[(c)] $\mathsf{A}_h \not\sigmam_{\rm PI} M_p$.
\end{enumerate}
\end{theo}
\betagin{proof}
Since $\mathsf{A}_h\subset \mathsf{A}_h(\mathbb{K})= \mathsf{A}_h \otimes_{\mathbb{F}}\mathbb{K}$ as $\mathbb{F}$-algebras, we can see that
$$\Id{\mathsf{A}_h \otimes_{\mathbb{F}}\mathbb{K}}\; = \; \IdK{\mathsf{A}_h(\mathbb{K})} \cap \mathbb{F}\lambdangle X\mathsf{R}A \; \subset \; \Id{\mathsf{A}_h}.$$
Part (b) of Theorem~\ref{Teorema Principal} concludes the proof of part (a). Part (b) follows from part (b) of Proposition \ref{Id(Ah) subset Id(Mp)}.
Consider $F_{p,q}(x,y)=G_{p,q}(x)\, R_{p,q}(x,y)\, (y^{q}-y)$ of $\mathbb{F}\lambdangle x,y\mathsf{R}A$, where
\betagin{align*}
G_{p,q}(x)&=(x^{q^{2}}-x)(x^{q^{3}}-x)\cdots(x^{q^{p}}-x),\\
R_{p,q}(x,y)&= \left(1-(y\,({\rm ad}\, x)^{p-1})^{q-1}\right)
\left(1-(y\,({\rm ad}\, x)^{p-2})^{q-1}\right) \cdots
\left(1-(y\,{\rm ad}\, x)^{q-1}\right)
\end{align*}
for $y\,{\rm ad}\,x=[y,x]$.
Genov~\cite{G. Genov} proved that $F_{p,q}(x,y)$ is a polynomial identity for $M_{p}$.
Since $x\,{\rm ad}\,x = [x,x] = 0$, for $x\in\mathsf{A}_h$ we have $R_{p,q}(x,x)=1$ and $$F_{p,q}(x,x)=(x^{q}-x)(x^{q^{2}}-x)(x^{q^{3}}-x)\cdots(x^{q^{p}}-x).$$
By part (b) of Lemma~\ref{lemma_embedding} elements $x,x^2,x^3,\ldots $ are linearly independent in $\mathsf{A}_h$. Therefore, $F_{p,q}(x,x)\neq0$ in $\mathsf{A}_h$; part (c) is proven.
\end{proof}
\betagin{conj}\lambdabel{conj}
$\Id{M_p(\mathbb{K})} \bigcap \mathbb{F}\lambdangle X\mathsf{R}A = \Id{\mathsf{A}_h}$.
\end{conj} | 1,064 | 28,609 | en |
train | 0.33.8 | \section{Counterexample}\lambdabel{section_example}
In this section we consider a counterexample to show that without Convention~\ref{conv1} Theorems~\ref{theo0} and~\ref{Teorema Principal} do not hold. Namely, we consider the commutative algebra $\mathsf{B}\sigmameq\mathbb{F}^2$ of double numbers from Example~\ref{ex2}, i.e., $\mathsf{B}$ has an $\mathbb{F}$-basis $\{1,\zeta\}$ with $\zeta^2=0$, and set $h=\zeta$. Note that Convention~\ref{conv1} does not hold for $h$. Then the statements of Theorems~\ref{theo0} and \ref{Teorema Principal} are not valid for $\mathsf{A}_{\zeta}(\mathbb{F}^2)=\mathsf{A}_h(\mathsf{B})$ (see Proposition~\ref{prop_ex} below).
\betagin{remark}\lambdabel{remark_ex}
If Convention~\ref{conv1} does not hold for $h$, then Lemmas~\ref{lemma_basis} and \ref{lemma_mult} are still valid for $\mathsf{A}_h(\mathsf{B})$.
\end{remark}
Part (b) of Lemma~\ref{lemma_mult} together with Remark~\ref{remark_ex} implies that for all $i,j,r,s\geq 0$ we have
\betagin{eq}\lambdabel{eq1_ex}
x^i \widehat{y}^j\cdot x^r \widehat{y}^s = x^{i+r}\, \widehat{y}^{s+j} + \zeta\, jr \, x^{i+r-1}\, \widehat{y}^{s+j-1},
\end{eq}
where we use conventions that $x^{-1}=0$ and $y^{-1}=0$. Then
\betagin{eq}\lambdabel{eq2_ex}
[x^i \widehat{y}^j, x^r \widehat{y}^s] = \zeta\, (jr-is) \, x^{i+r-1}\, \widehat{y}^{s+j-1} \;\text{ in }\; \mathsf{A}_{\zeta}(\mathbb{F}^2).
\end{eq}
The unital finite dimensional Grassmann algebra $\mathsf{G}_k$ of rank $k$ has an $\mathbb{F}$-basis
$$\{1,e_{i_1}\cdots e_{i_m} \ | \ 1\leq i_1<\cdots < i_m\leq k\}$$
and satisfies the defining relations $e_i^2=0$ and $e_i e_j = - e_j e_i$ for all $1\leq i,j\leq k$. The polynomial identities for $\mathsf{G}_k$ were described by Di Vincenzo~\cite{DiVincenzo_1991} for $p=0$ and by Giambruno, Koshlukov~\cite{GiambrunoKoshlukov_2001} for any infinite field.
\betagin{prop}\lambdabel{prop_ex} Assume that $\mathbb{F}$ is an infinite field.
\betagin{enumerate}
\item[(a)] The T-ideal of identities $\Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$ is generated by $$ f_1=[[x_1,x_2],x_3],\quad f_2=[x_1,x_2]\, [x_3,x_4].$$
\item[(b)] $\mathsf{A}_{\zeta}(\mathbb{F}^2)\not\sigmam_{\rm PI} M_t(\mathsf{C})$ for every $t\geq 2$ and every $\mathbb{F}$-algebra $\mathsf{C}$ with unity.
\item[(c)] $\mathsf{A}_{\zeta}(\mathbb{F}^2)\sigmam_{\rm PI} \mathsf{G}_k$ if and only if $k\in\{2,3\}$.
\end{enumerate}
\end{prop}
\betagin{proof}
\noindent{\bf (a)} By $\mathbb{F}$-linearity formula~\mathsf{R}ef{eq2_ex} implies that $[a,b]$ belongs to $\zeta\, \mathsf{A}_{\zeta}(\mathbb{F}^2)$ for all $a,b\in \mathsf{A}_{\zeta}(\mathbb{F}^2)$. Then $f_1,f_2\in \mathbb{F}\lambdangle X\mathsf{R}A$ are nontrivial polynomial identities for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$, since $\zeta^2=0$. Note that
$$f_3 = [x_1,x_2]\,x_3\, [x_4,x_5] = [[x_1,x_2],x_3] [x_4,x_5] + x_3 [x_1,x_2] [x_4,x_5]\in \Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$$
follows from $f_1,f_2$. Denote by $I$ the T-ideal generated by $f_1,f_2$.
Assume that $f=\sum_k \alpha_k w_k$ is a nontrivial identity for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$, where $\alpha_k\in\mathbb{F}$ and $w_k\in\mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ is a monomial. Since $\mathbb{F}$ is infinite, we can assume that $f$ is multihomogeneous. i.e., there exists $\un{d}\in\mathbb{N}^m$ with $\mathop{\rm mdeg}(w_k)=\un{d}$ for each $k$. We apply equalities
$$u x_j x_i v = u x_i x_j v - u [x_i,x_j] v,$$
$$u [x_i,x_j] v = [x_i,x_j] u v - [[x_i,x_j], u] v,$$
where monomials $u,v$ can be empty and $i<j$, to monomials $\{w_k\}$ and then repeat this procedure. Since $f_1,f_3\in \Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$, we finally obtain that there exist $g\in I$, $\alpha_0, \alpha_{ij}\in\mathbb{F}$ such that
$$f = g + \alpha_0 x_1^{d_1}\cdots x_m^{d_m} + \sum_{1\leq i<j\leq m} \alpha_{ij} [x_i,x_j] x_1^{d_1}\cdots x_i^{d_i-1} \cdots x_j^{d_j-1} \cdots x_m^{d_m} \;\text{ in }\;\mathbb{F}\lambdangle X\mathsf{R}A.$$
Since $f(1,\ldots,1) = g(1,\ldots,1) + \alpha_0$, we obtain that $\alpha_0=0$. Consider $i<j$ with $d_i,d_j\geq1$. Making substitutions $x_i\to x$, $x_j\to \widehat{y}$, $x_l\to 1$ for each $l$ different from $i$ and $j$, we can see that $0=0 + \alpha_{ij} [x,\widehat{y}] x^{d_i-1}\widehat{y}^{d_j-1}$ in $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. Thus $-\alpha_{ij} \zeta x^{d_i-1}\widehat{y}^{d_j-1}=0$ in $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. Lemma~\ref{lemma_basis} together with Remark~\ref{remark_ex} implies that $\alpha_{ij}=0$. Therefore, $f=g$ lies in $I$.
\noindent{\bf (b)} Since $\mathbb{F}\subset \mathsf{C}$, every polynomial identity for $M_t(\mathsf{C})$ lies in $\Id{M_t(\mathbb{F})}$. By Amitsur--Levitzki Theorem~\cite{Amitsur_Levitzki} the minimal degree of a polynomial identity for $M_t(\mathbb{F})$ is $2t$. In particular, $f_1$ is not an identity for $M_t(\mathsf{C})$.
\noindent{\bf (c)} Since $\mathsf{G}_k$ is commutative in case $p=2$ or $k=1$, we can assume that $p\neq 2$ and $k\geq2$. Note that
\betagin{eq}\lambdabel{eq3_ex}
f_2(e_1,e_2,e_3,e_4)=4e_1 e_2 e_3 e_4 \neq 0 \;\text{ in }\; \mathsf{G}_k \;\text{ for }\; k\geq4.
\end{eq}
Thus we can assume that $k\in\{2,3\}$. The T-ideal $\Id{\mathsf{G}_k}$ is generated by
\betagin{enumerate}
\item[$\bullet$] $f_1,f_2$ in case $p=0$ or $p=k=3$.
\item[$\bullet$] $f_1,{\rm St}_4$ in case $p>k$, where $k\in\{2,3\}$.
\end{enumerate}
Since
$${\rm St}_4(x_1,x_2,x_3,x_4) = [x_1,x_2]\circ [x_3,x_4] - [x_1,x_3]\circ [x_2,x_4] + [x_1,x_4] \circ [x_2,x_3],$$
where $u\circ v$ stands for $uv+vu$, part (a) implies that ${\rm St}_4$ lies in $\Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$. On the other hand, we can see that $f_2$ is a polynomial identity for $\mathsf{G}_k$ when $k\in\{2,3\}$. Part (c) is proven.
\end{proof} | 2,496 | 28,609 | en |
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\end{document}
\appendix | 1,881 | 28,609 | en |
train | 0.33.10 | \section{Some remarks and proofs}
\betagin{remark}\lambdabel{remark0} Assume $p=0$ and $h\in \mathbb{F}^{*}$.
The classical proof of the fact that $\mathsf{A}_1$ does not have nontrivial polynomial identities is also valid for $\mathsf{A}_h$. Namely, Kaplansky's Theorem from
\betagin{enumerate}
\item[$\bullet$] I. Kaplansky, {\it Rings with a polynomial identity}, Bull. Amer. Math. Soc. {\bf 54} (1948), 575--580.
\end{enumerate}
states that any primitive $\mathbb{F}$-algebra $A$ satisfying a polynomial identity of degree $d>0$ is a simple central algebra of finite dimension and $\dim_{\mathrm{Z}(A)}A\leq[d/2]^{2}$.
Since the algebra $\mathsf{A}_h$ is simple by part (d) of Proposition \ref{Obs Prop Ah} and unitary, then it is also primitive. Moreover, ${\rm Z}(\mathsf{A}_h)=\mathbb{F}$ by part~(b) of Proposition \ref{Obs Prop Ah}. Assume that $\mathsf{A}_h$ has a nontrivial polynomial identity. Then Kaplansky's theorem implies that the dimension of $\mathsf{A}_h$ over $\mathbb{F}={\rm Z}(\mathsf{A}_h)$ is finite; a contradiction.
\end{remark}
\betagin{remark}
The matrices $A_0$ and $B_0$ of $M_p$ defined in Section~\ref{section_positive} are the following matrices:
$$A_{0}=\betagin{pmatrix}
0&0&\cdots&0&0\\
1&0&\cdots&0&0\\
0&1&\cdots&0&0\\
\vdots&\vdots&&\vdots&\vdots\\
0&0&\cdots&1&0
\end{pmatrix}\;\;\text{and}\;\; B_{0}=\betagin{pmatrix}
0&1&0&\cdots&0\\
0&0&2&\cdots&0\\
\vdots&\vdots&\vdots&&\vdots\\
0&0&0&\cdots&p-1\\
0&0&0&\cdots&0
\end{pmatrix}.$$
\end{remark}
\betagin{proof_of}{of Lemma~\ref{lemma_A0_B0}.}
\noindent{\bf (a)} [formula for $A_0^k$]
For $k = 1$ the statement holds by the definition of $A_{0}$. Suppose the statement holds for $k<p-1$. Then
\betagin{align*}
A_{0}^{k+1}&=\left(\sum_{j=1}^{p-k}E_{(j+k)j}\right)\left(\sum_{r=1}^{p-1}E_{(r+1)r} \right)=\sum_{j=1}^{p-k}E_{(j+k)(j-1)}=\sum_{i=1}^{p-(k+1)}E_{(i+k+1)i}.
\end{align*}
The claim is proven.
\noindent{\bf (b)}
\betagin{align*}
B_{0}A_{0}-A_{0}B_{0}&=\left(\sum_{i=1}^{p-1} iE_{i(i+1)} \right) \left(\sum_{j=1}^{p-1}E_{(j+1)j} \right)-\left(\sum_{j=1}^{p-1} E_{(j+1)j} \right) \left(\sum_{i=1}^{p-1} i E_{i(i+1)}\right)\\
&=\sum_{i=1}^{p-1} i E_{ii}-\sum_{i=1}^{p-1} i E_{(i+1)(i+1)}\\
&=I_{p}
\end{align*}
\end{proof_of}
\betagin{proof_of}{of Corollary~\ref{cor_D0}.}
By Lemma~\ref{lemma_hA} and the definition of $D_0$ we have
$$D_{0} = B_{0}\varepsilon_{\alpha}(h(A)) = \left(\sum_{r=1}^{p-1}r\, E_{r(r+1)} \right)
\left( \sum_{i=1}^{p}\sum_{j=1}^{i} \frac{1}{(i-j)!} h^{(i-j)}(\alpha) E_{ij} \right). $$
The required follows.
\end{proof_of}
\betagin{lemma}\lambdabel{base para Mp}
The set $\betata=\{A_{0}^{r}B_{0}^{s} \ | \ 0\leq r,s < p\}$ is a base for $M_{p}(\mathbb{F})$.
\end{lemma}
\betagin{proof}
Since $|\betata|=p^{2}=\dim_{\mathbb{F}}{M_{p}(\mathbb{F})}$ then it is sufficient to prove that $\betata $ is a linearly independent set.
Suppose that there are scalars, not all zero, $a_{rs}\in \mathbb{F}$ with $0\leq r,s<p$ such that: $$\sum_{0\leq r,s<p}a_{rs}A_{0}^{r}B_{0}^{s}=0.$$
Let $s_{0}=\min\{s \ | \ a_{rs}\neq 0 \ \ \text{for some} \ \ 0\leq r<p\}$ , then we have
\betagin{align*}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{s_{0}}+\sum_{\substack{0\leq r<p\\ s_{0}<s<p}}a_{rs}A_{0}^{r}B_{0}^{s}.\\
\intertext{Since $0\leq s_{0}<s<p$ then $0\leq p-s_{0}-1<p$ and $s+p-s_{0}-1>p-1$ hence, by parts (b) and (c) of Lemma~\ref{lemma_A0_B0}, $B_{0}^{s+p-s_{0}-1}=0$ e $B^{p-s_{0}-1}\neq 0$. thus}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}+\sum_{\substack{0\leq r<p\\ s_{0}<s<p}}a_{rs}A_{0}^{r}B_{0}^{s+p-s_{0}-1},\\
\intertext{then}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}.
\end{align*}
Let $r_{0}=\max\{r \ | \ a_{rs_{0}\neq 0}\}$, then is immediate $a_{r_{0}s_{0}}\neq 0$. Hence
\betagin{align*}
0&=a_{r_{0}s_{0}}A_{0}^{r_{0}}B_{0}^{p-1}+\sum_{0\leq r<r_{0}}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}.\\
\intertext{As $r_{0}\geq 0$ then $p-r_{0}-1<p$, hence $A_{0}^{p-r_{0}-1}\neq 0.$ then}
0&=a_{r_{0}s_{0}}A_{0}^{p-1}B_{0}^{p-1}+\sum_{0\leq r<r_{0}}a_{rs_{0}}A_{0}^{p-r_{0}-1+r}B_{0}^{p-1}.\\
\intertext{Since $r<r_{0}$ then $p-r_{0}-1+r<p-1$, this implies that $A_{0}^{p-r_{0}-1+r}\neq A_{0}^{p-1}$. Hence}
0&= a_{r_{0}s_{0}}A_{0}^{p-1}B_{0}^{p-1}=a_{r_{0}s_{0}}(p-1)!E_{pp.}
\end{align*}
From where $a_{r_{0}s_{0}}(p-1)!=0$. Since $(p-1)!\neq 0$ module $p$, we have $a_{r_{0}s_{0}}=0$. this is a contradiction. Therefore, $\betata$ is a base for $M_{p}(\mathbb{F})$.
\end{proof}
\betagin{prop}\lambdabel{Obs Prop Ah}
\betagin{enumerate}
\item[(a)] (\cite{Benkart_Lopes_Ondrus_I}, Theorem 5.3) If $p=0$, then the center ${\rm Z}(\mathsf{A}_h)$ of $\mathsf{A}_h$ is $\mathbb{F}$; if $p>0$, then ${\rm Z}(\mathsf{A}_h)=\mathbb{F}[x^{p},h^{p}y^{p}]$.
\item[(b)] (\cite{Benkart_Lopes_Ondrus_I}, Proposition 5.5) If $p>0$, then $\mathsf{A}_h$ is a free module over ${\rm Z}(\mathsf{A}_h)$ and the set $\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$ is a basis.
\item[(c)] (\cite{Benkart_Lopes_Ondrus_I}, Corollary 7.4) The algebra $\mathsf{A}_h$ is simple if and only if $p=0$ and $h\in \mathbb{F}^{*}$.
\end{enumerate}
\end{prop}
\section{Some references}
\betagin{enumerate}
\item[(1)]
Kaplansky's Theorem (see Remark~\ref{remark0}) can be found in
\betagin{enumerate}
\item[$\diamond$] I.N. Herstein, {\it Noncommutative rings}, The Carus Math. Monographs 15, 1973 (see Theorem 6.3.2).
\item[$\diamond$] \cite{A.Gia} (see Theorem 1.11.7, page 31).
\end{enumerate}
\item[(2)] [Remark for the proof of Corollary~\ref{Id(Mp) subset Id(Ah)}].
If $R$ is an algebra over a commutative ring $C$, then $M_{n}(R)\cong R\otimes_{C}M_{n}(C)$. This remark can be found in
\betagin{enumerate}
\item[$\diamond$]L.H. Rowen, {\it Polynomial Identities in Rings Theory}, Academic Press, New York, 1980 (see Proposition 1.8.15)).
\end{enumerate}
\item[(3)] Assume that $\mathbb{F}$ is a finite field of order $q=p^{k}$. Then in
\betagin{enumerate}
\item[$\diamond$] Yu. N. Maltsev, E.N. Kuzmin, {\it A basis for identities of the algebra of second order matrices over a finite field}, Algebra Logic 17 (1978), 17-21.
\end{enumerate}
it was proved that $\Id{M_{2}(\mathbb{F})}$ is generated by
\betagin{align*}
f_{1}(x,y)&=(x-x^q)(y-y^{q^{2}})(1-[x,y]^{q-1}),\\
f_{2}(x,y)&=(x-x^q)\circ(y-y^{q^{2}})[(x-x^q)\circ(y-y^{q})]^{q},
\end{align*}
where $[x,y]=xy-yx$, $x\circ y=xy+yx$.
\end{enumerate} | 2,999 | 28,609 | en |
train | 0.33.11 | \section{Some references}
\betagin{enumerate}
\item[(1)]
Kaplansky's Theorem (see Remark~\ref{remark0}) can be found in
\betagin{enumerate}
\item[$\diamond$] I.N. Herstein, {\it Noncommutative rings}, The Carus Math. Monographs 15, 1973 (see Theorem 6.3.2).
\item[$\diamond$] \cite{A.Gia} (see Theorem 1.11.7, page 31).
\end{enumerate}
\item[(2)] [Remark for the proof of Corollary~\ref{Id(Mp) subset Id(Ah)}].
If $R$ is an algebra over a commutative ring $C$, then $M_{n}(R)\cong R\otimes_{C}M_{n}(C)$. This remark can be found in
\betagin{enumerate}
\item[$\diamond$]L.H. Rowen, {\it Polynomial Identities in Rings Theory}, Academic Press, New York, 1980 (see Proposition 1.8.15)).
\end{enumerate}
\item[(3)] Assume that $\mathbb{F}$ is a finite field of order $q=p^{k}$. Then in
\betagin{enumerate}
\item[$\diamond$] Yu. N. Maltsev, E.N. Kuzmin, {\it A basis for identities of the algebra of second order matrices over a finite field}, Algebra Logic 17 (1978), 17-21.
\end{enumerate}
it was proved that $\Id{M_{2}(\mathbb{F})}$ is generated by
\betagin{align*}
f_{1}(x,y)&=(x-x^q)(y-y^{q^{2}})(1-[x,y]^{q-1}),\\
f_{2}(x,y)&=(x-x^q)\circ(y-y^{q^{2}})[(x-x^q)\circ(y-y^{q})]^{q},
\end{align*}
where $[x,y]=xy-yx$, $x\circ y=xy+yx$.
\end{enumerate}
\section{We do not need it!}
A noncommutative polynomial $f(x_{1},\ldots,x_{n})=\sum\alphapha_{i}x_{i_{1}}\cdots x_{i_{d}}$ of $\mathbb{F}\lambdangle x_1,\ldots,x_n\mathsf{R}A$ is called homogeneous of degree $d$, where $\alpha_i\in\mathbb{F}$. $f(x_{1},\ldots,x_{n})$ is homogeneous of multidegree $(d_{1},\ldots,d_{n})$ if each variable $x_{i}$ appears the same number of times $d_{i}$ in all monomials.
The polynomial $f(x_{1},\ldots,x_{n})$ is multilinear of degree $n$ if it homogeneous of multidegree $(1,\ldots,1$) ($n$ times), i.e.,
$$f(x_{1},\ldots,x_{n})=\sum_{\sigmagma\in S_{n}}\alphapha_{\sigmagma}x_{\sigmagma(1)}\cdots x_{\sigmagma(n)}$$ for some $\alphapha_{\sigmagma}\in\mathbb{F}$.
\betagin{remark}\lambdabel{Obs Prop Basicas PI} The following statements are well known.
\betagin{enumerate}
\item[a)] Let $A$ be an algebra over a field $\mathbb{F}$ spanned by a set $\betata$. It is easy to check that if a multilinear polynomial $f(x_{1},\cdots,x_{n})$ vanished on $\betata$, then $f(x_{1},\cdots,x_{n})$ is a polynomial identity for $A$.
\end{enumerate}
\end{remark}
Part~(c) of Proposition~\ref{Obs Prop Ah} together with Corollary 1.19 from
\betagin{enumerate}
\item[$\bullet$] V. Drensky, E. Formanek, {\it Polynomial identity rings}, CRM Barcelona, Springer Basel AG, 2004.
\end{enumerate}
implies that in case $p>0$ the algebra $\mathsf{A}_h$ have nontrivial polynomial identities.
The following result provides an important tool for study multilinear polynomial identities for the algebra $\mathsf{A}_h$.
\betagin{lemma}\lambdabel{Lema pol multilineal}
Assume $p>0$ and $f\in\FF\lambdangle X\rangle$ is a multilinear polynomial. Then $f$ is a polynomial identity for $\mathsf{A}_h$ if and only if $ f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}$ from $S=\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$.
\end{lemma}
\betagin{proof}
If $f$ is polynomial identity for $\mathsf{A}_h$, then $f(b_{1},\ldots,b_{n})=0$ for all $b_{1},\ldots,b_{n}\in \mathsf{A}_h$, in particularly $ f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in S=\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$, because $S\subset \mathsf{A}_h$. Reciprocally, from part (a) of Proposition \ref{Obs Prop Ah} it is clear that $\betata=\{x^{i}h^{j}y^{j} \ | \ i,j\geq 0\}$ is a generator set of $\mathsf{A}_h$, and since $f$ is multilinear polynomial it is sufficient to prove that $f$ vanish on $\betata$ (see Remark \ref{Obs Prop Basicas PI}, item $a)$). Let $b_{1},\ldots,b_{n}\in \betata$, then as the center of $\mathsf{A}_h$ is $\mathbb{F}[x^{p},h^{p}y^{p}]$ we have that $b_{k}=c_{k}w_{k}$, where $c_{k}\in {\rm Z}(\mathsf{A}_h)$ and $w_{k}\in S$ for all $1\leq k\leq n$. Hence, $$f(b_{1},\ldots,b_{n})=f(c_{1}w_{1},\ldots,c_{n}w_{n})=c_{1}\cdots c_{n}f(w_{1},\ldots,w_{n})=0,$$ because $f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in S$. Therefore, $f$ is polynomial identity for $\mathsf{A}_h$.
\end{proof}
The well-known Amitsur theorem (see \cite{A.Gia}, Theorem 1.7.7) states that the standard polynomial ${\rm St}_{2n}$ is a polynomial identity for $M_{n}$, the $n\times n$-matrix algebra over $\mathbb{F}$. The following example proves that the standard polynomial of degree $4$, ${\rm St}_{4}$, is a polynomial identity for $\mathsf{A}_h$ if the characteristic of $\mathbb{F}$ is $2$.
\betagin{example}
If the characteristic of $\mathbb{F}$ is 2, then $\mathsf{A}_h$ satisfies the standard identity ${\rm St}_{4}=0$.
\end{example}
\betagin{proof}
As ${\rm St}_{4}$ is a multilinear polynomial, by Lemma \ref{Lema pol multilineal}, it is sufficient to prove that ${\rm St}_{4}$ vanish on $S=\{1,x,hy,xhy\}$. Let $a_{1},a_{2},a_{3},a_{4}\in S$. Suppose $a_{i}=a_{j}$ for some $i\neq j$ then for each even permutation $\sigmagma\in S_{4}$ there is an odd permutation $\tau=\sigmagma\cdot(i \ j)$ such that $a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=a_{\tau(1)}a_{\tau(2)}a_{\tau(3)}a_{\tau(4)}$, hence \[{\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=2\sum_{\sigmagma\ par}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}.\] Then, ${\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=0$, because the characteristic of $\mathbb{F}$ is $2$. On the other hand, if $a_{i}\neq a_{j}$ for all $i\neq j$, then for each permutation $\sigmagma\in S_{4}$ there is $1\leq i\leq 4$ such that $a_{\sigmagma(i)}=1$, hence \[{\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=4\sum_{\pi\in S_{3}} (-1)^{\pi}b_{\pi(1)}b_{\pi(2)}b_{\pi(3)},\] where $b_{1},b_{2},b_{3}\in \{x,hy,xhy\}$. As $\mathbb{F}$ has characteristic 2, ${\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=0$. Therefore, ${\rm St}_{4}$ is a polynomial identity for $\mathsf{A}_h$.
\end{proof}
\section{Open question: Is a polynomial $h(x)$ over an algebra with non divisor-zero coefficient of the highest term satisfies $h(\alpha)$ is not a divisor-zero for some $\alpha\in \mathbb{F}$? }
\noindent{\bf Question. }{\it
Assume that $\mathsf{B}$ is a unital algebra (maybe non-commutative) over an infinite field $\mathbb{F}$ and $h$ is a polynomial from $\mathsf{B}[x]$ with all coefficients from the center $Z(\mathsf{B})$.
Moreover, the coefficient of the highest term of $h$ is not a zero-divisor.
Is it true that $h(\alpha)$ is not a zero-divisor for some $\alpha\in Z(\mathsf{B})$?
}
\end{document} | 2,727 | 28,609 | en |
train | 0.34.0 | \begin{document}
\title{A Near Optimal Approximation Algorithm for Vertex-Cover Problem}
\author{Deepak Puthal\footnote{[email protected]} \\
Department of Computer Science \& Engineering\\
National Institute of Technology Silchar\\
Silchar, Assam, India}
\maketitle
\begin{abstract}
Recently, there has been increasing interest and progress in improvising the approximation algorithm for well-known NP-Complete problems, particularly the approximation algorithm for the Vertex-Cover problem. Here we have proposed a polynomial time efficient algorithm for vertex-cover problem for more approximate to the optimal solution, which lead to the worst time complexity $\Theta(V^2)$ and space complexity $\Theta(V+E)$. We show that our proposed method is more approximate with example and theorem proof. Our algorithm also induces improvement on previous algorithms for the independent set problem on graphs of small and high degree.
\end{abstract}
{\bf Keywords}:Approximation algorithm \and Vertex-Cover Problem \and Complexity \and Adjacency list.
\section{Introduction}
\label{intro}
A graph G represents as G = (V, E): V is number of vertices in graph and E is number of edges in graph and impliment as an adjacency lists or as an adjacency matrix for both directed and undirected graphs. There are two types of graph i.e. (I) Sparse graphs-those for which $|E|$ is much less than $|V|^2$ $(E<<V^2)$. (II) Dense graphs-those for which $|E|$ is close to $|V|^2 (E \simeq V^2)$. Here we presented the graph as the adjacency-list for the evaluation of our algorithm. \\
The vertex-cover problem is to find a minimum number of vertex to cover a given undirected graph. We call such a vertex cover an optimal vertex cover. This problem is the optimization version of an NP-complete decision problem. Proposed algorithm is polynomial time algorithm in order to find the set of vertex to cover the graph. Which shows the better performance than the traditional algorithm for vertex cover \cite{RefB1}.
\section{The vertex-cover problem}
As it is NP-Hard problem so it is hard to find an optimal solution of a graph G, but not difficult to find a near optimal solution. Our propose method gives very near optimal solution for Vertex-cover problem. The following approximation algorithm takes an undirected graph G as input \cite{RefB1} and returns a set of vertex to cover the graph and whose size is less than the previous method. \\
All graphs mentioned here are simple undirected graph. We follow \cite{RefB2} for definitions. Our proposed method (See algorithm and Fig.\ref{fig:2}) is on undirected graph. Here we used the adjacency list to represent graph G. We introduce a new field \textit{weight} in the \textit{list} to store the degree of each individual vertex. \textit{i.e.}
struct list
\{
\hspace{5 mm} char vertex;
\hspace{5 mm} int weight;
\hspace{5 mm} struct node *next;
\hspace{5 mm} struct node *ref;
\};
\begin{algorithm}
\caption{Approximate Vertex-Cover Algorithm}
\begin{algorithmic} [1]
\REQUIRE {{In the \textit{List} we introduce another field \textit{weight} \\ The value of \textit{weight} is number of node in reference \textit{(ref)}}}
\STATE $C^+ \leftarrow \emptyset $
\STATE L = \textit{List}
\STATE L[w] = Reference weight
\STATE (h, v) = highest weight of the list and respective vertex \label{marker}
\IF {$h \neq 0$}
\STATE $C^+ \leftarrow C^+ \cup {v}$
\STATE v[w] $\leftarrow 0$
\FOR{all vertex of \textit{List} L[ref] $\in \{v\}$}
\STATE L[w] $\leftarrow$ L[w]-1
\ENDFOR
\STATE \textbf{go to} \ref{marker}
\ELSE
\RETURN $C^+$
\ENDIF
\end{algorithmic}
\end{algorithm}
It's space complexity is $\Theta(V+E)$ \cite{RefB1}. For step 4 search in the list is O(V). In the for loop \textit{i.e. step 8 to 10} for each individual vertex need to search its reference vertices. So worst time complexity is $O(V*(V-1))$= $O(V^2)$. So the worst time complexity of the graph is $\Theta(V^2)$.
\newtheorem{mydef}{Theorem}
\begin{mydef}
\emph{(Thomas H. Cormen et.al. \cite{RefB1})}
\label{vcp1}
APPROX-VERTEX-COVER is a polynomial-time 2-approximation algorithm.
\end{mydef}
\begin{mydef}
\label{vcp2}
Proposed Approximate Vertex-Cover is a polynomial-time $(2-\varepsilon)$-approximation algorithm.
\end{mydef}
\begin{proof}
In Theorem ~\ref{vcp1} $C$ is the set of vertex for APPROX-VERTEX-COVER and $C^*$ in the optimal vertex cover $i.e. |C| \leq 2|C^*|$. In our approach we pick one vertex and remove the edges connected to that vertex. So most of the times we don't consider both end point of one edges, which followed in Theorem ~\ref{vcp1}. For our proposed method we consider the resultant set of vertex is $C^+$, then $|C^+| = |C| - \varepsilon$.\\
$ \Rightarrow |C^+| = (2- \varepsilon) |C^*|$, $0 \leq \varepsilon \leq 1$.
\qed
\end{proof}
In some cases proposed method$(C^+)$ approaches to optimal solution when $\varepsilon$ value is 1. Our method is shown in Fig. \ref{fig:1}, algorithm (Approximate Vertex-Cover), and proved in Theorem ~\ref{vcp2}. The comparison of the optimal vertex-cover, previous vertex-cover and proposed vertex-cover shown in Fig. \ref{fig:2}.
\begin{figure}
\caption{The operation of Approximate Vertex-Cover. \textit{(a)}
\label{fig:1}
\end{figure}
\begin{figure*}
\caption{Comparison of optimal, Traditional and proposed vertex cover result. \textit{(a)}
\label{fig:2}
\end{figure*}
\section{Conclusion}
Here in our proposed technique we produce the set of vertex for vertex-cover problem. Which is more near optimal solution and better than the previous technique.
\end{document} | 1,668 | 1,668 | en |
train | 0.35.0 | \begin{document}
\author{Henrik Schlichtkrull}
\address[Henrik Schlichtkrull]{Department of Mathematics, University of
Copenhagen\\ Universitetsparken 5, DK-2100 Copenhagen \O}
\email{[email protected]}
\author{Peter Trapa}\thanks{The second author was supported in part by
NSF grant DMS-1302237.}
\address[Peter Trapa]{Department of Mathematics, University of Utah\\
Salt Lake City, UT 84112}
\email{[email protected]}
\author{David A. Vogan, Jr.}
\address[David Vogan]{Department of Mathematics, MIT \\ Cambridge, MA 02139}
\email{[email protected]}
\title{Laplacians on spheres}
\date{\today}
\maketitle
\begin{abstract}
Spheres can be written as homogeneous spaces $G/H$ for compact Lie
groups in a small number of ways. In each case, the decomposition of
$L^2(G/H)$ into irreducible representations of $G$ contains
interesting information. We recall these decompositions, and see what
they can reveal about the analogous problem for noncompact real forms
of $G$ and $H$.
\end{abstract}
\noindent This paper is dedicated to Joe Wolf, in honor of all that we
have learned from him about the connections among geometry,
representation theory, and harmonic analysis; and in gratitude for
wonderful years of friendship.
\section{Introduction}\label{sec:intro}
\setcounter{equation}{0}
The sphere has a Riemannian metric, unique up to a positive scale,
that is preserved by the action of the orthogonal group. Computing the
spectrum of the Laplace operator is a standard and beautiful
application of representation theory. These notes will look at some
variants of this computation, related to interesting subgroups of the
orthogonal group.
The four variants presented in Sections \ref{sec:R}, \ref{sec:C},
\ref{sec:H}, and \ref{sec:O} correspond to the following very general
fact, due to \'Elie Cartan: if $G/K$ is an irreducible compact
Riemannian symmetric space of real rank $1$, then $K$ is
transitive on the unit sphere in $T_{eK}(G/K)$. (The only caveat is in
the case of the one-dimensional symmetric space ${S^1}$. In this case
one needs to use the full isometry group $O(2)$ rather than its
identity component to get the transitivity.) The isotropy group of
a point on the sphere is often called $M$ in the theory; so the
conclusion is that
\begin{equation}\label{e:rank1}
\text{sphere of dimension ($\dim G/K -1$)} \simeq K/M
\end{equation}
The calculations we do correspond to the rank one symmetric spaces
\begin{alignat*}{5}
&O(n+1)/O(n), && S^{n-1} \simeq O(n)/O(n-1) \qquad &&\text{(Section
\ref{sec:R}),}\\
&SU(n+1)/U(n),&& S^{2n-1} \simeq U(n)/U(n-1) &&\text{(Section \ref{sec:C}),}\\[.5ex]
&Sp(n+1)/[Sp(n)\times Sp(1)],\quad &&\begin{matrix} S^{4n-1} \simeq
[Sp(n)\times Sp(1)]/ \\[.2ex] \qquad [Sp(n-1)\times
Sp(1)_\Delta] \end{matrix} &&\text{(Section
\ref{sec:H}), and}\\[.5ex]
&F_4/\Spin(9) && S^{15} \simeq \Spin(9)/\Spin(7)' &&\text{(Section
\ref{sec:O}).}
\end{alignat*}
The representations of $O(n)$, $U(n)$, $Sp(n)\times Sp(1)$, and
$\Spin(9)$ that we are computing are exactly the $K$-types of the
spherical principal series representations for the noncompact forms of
the symmetric spaces.
Rank one symmetric spaces provide three infinite families (and one
exceptional example) of realizations of spheres as homogeneous spaces
(for compact Lie groups). A theorem due to Montgomery-Samelson and
Borel (\cite{MS} and \cite{BOct}; there is a nice account in
\cite{Wolf}*{(11.3.17)}) classifies {\em all} such realizations. In
addition to some minor variants on those above, like
$$S^{2n-1} \simeq SU(n)/SU(n-1),\qquad S^{4n-1} \simeq
Sp(n)/Sp(n-1),$$
the only remaining possibilities are
$$\begin{aligned}
S^6 \simeq G_{2,c}/SU(3) \qquad &\text{(Section \ref{sec:G2}), and}\\
S^7 \simeq \Spin(7)/G_{2,c} \qquad &\text{(Section \ref{sec:bigG2}).}\\
\end{aligned}$$
After recalling in Sections \ref{sec:R}--\ref{sec:bigG2} the classical
harmonic analysis related to these various realizations of spheres, we
will examine in Sections \ref{sec:invt}--\ref{sec:size} what these
classical results say about invariant differential operators.
In Sections \ref{sec:Opq}--\ref{sec:bigncG2} we examine what
this information about harmonic analysis on spheres can tell us about
harmonic analysis on hyperboloids. With $n=p+q$ the symmetric spaces
$$H_{p,q}=O(p,q)/O(p-1,q),\quad (0\le q\le n)$$
are said to be {\it real forms} of each other (and thus in
particular of
$$S^{n-1}=O(n)/O(n-1) = H_{n,0}).$$
Similarly, each of the realizations listed above of $S^{n-1}$ as a
non-symmetric homogeneous space for a subgroup of $O(n)$ corresponds
to one or more noncompact real forms, realizing some of the $H_{p,q}$
as non-symmetric homogeneous spaces for subgroups of $O(p,q)$. These
realizations exhibit the hyperbolic spaces as examples of real
spherical spaces of rank one, and as such our interest is primarily
with their discrete series. These and related spaces have previously been
studied by T.~Kobayashi (see \cite{Kobayashi-Stiefel,Kob,toshi:zuckerman,toshi:howe}). In Sections
\ref{sec:Upq}--\ref{sec:bigncG2} we give an essentially self-contained treatment,
in some cases giving slightly more refined information. In particular, we obtain some
interesting discrete series representations for small parameter values
for the real forms of
$S^6 \simeq G_{2,c}/SU(3)$.
\begin{subequations}\label{se:orbitmethod}
For information about real spherical spaces and their discrete series
in general we refer to \cite{KKOS}; this paper was intended in part to
examine some interesting examples of those results. In particular, we
are interested in formulating the parametrization of discrete series
in a way that may generalize as much as possible. We are very grateful
to Job Kuit for extensive discussions of this parametrization problem.
One such formulation involves the ``method of coadjoint orbits:''
representations of $G$ are parametrized by certain orbits $G\cdot
\lambda$ of $G$ on the real dual vector space
\begin{equation}\label{eq:coadjoint}
{\mathfrak g}_0^* =_{\text{def}} \Hom_{\mathbb R}(\Lie(G),{\mathbb R})
\end{equation}
(often together with additional data). The orbits corresponding to
representations appearing in $G/H$
typically have representatives
\begin{equation}\label{eq:GmodH}
\lambda \in [{\mathfrak g}_0/{\mathfrak h}_0]^*.
\end{equation}
We mention this at the beginning of the paper because this coadjoint
orbit parametrization is often {\em not} a familiar one (like that of
representations of compact groups by highest weights). We will write
something like
\begin{equation}\label{eq:orbitparam}
\pi(\text{orbit\ }\lambda,\Lambda)
\end{equation}
for the representation of $G$ parametrized by $G\cdot\lambda$ (and
sometimes additional data $\Lambda$). If $G$ is an equal-rank reductive group
and $\lambda \in {\mathfrak g}_0^*$ is a regular elliptic element
(never mind exactly what these terms mean), then
\begin{equation}\label{eq:dsparam}
\pi(\text{orbit\ }\lambda) = \text{discrete series with
Harish-Chandra parameter $i\lambda$;}
\end{equation}
so this looks like a moderately familiar parametrization. (Here
``discrete series representation'' has the classical meaning of an
irreducible summand of $L^2(G)$. Soon we will use the term more
generally to refer to summands of $L^2(G/H)$.) But notice
that \eqref{eq:dsparam} includes the case of $G$ compact. In that case
$\lambda$ is not the highest weight, but rather an exponent in the
Weyl character formula.
Here is how most of our discrete series will arise. Still for $G$
reductive, if $\lambda$ is elliptic but possibly singular, define
\begin{equation}\label{eq:cohindA}
G^\lambda = L, \qquad {\mathfrak q} = {\mathfrak l} + {\mathfrak u}
\end{equation}
to be the $\theta$-stable parabolic subalgebra defined by the
requirement that
\begin{equation}\label{eq:cohindB}
i\lambda(\alpha^\vee) > 0, \qquad (\alpha \in \Delta({\mathfrak
u},{\mathfrak h})).
\end{equation}
The ``additional data'' that we sometimes need is a one-dimensional character
\begin{equation}\label{eq:cohindC}
\Lambda \colon L \rightarrow {\mathbb C}^\times, \quad d\Lambda =
i\lambda + \rho({\mathfrak u}).
\end{equation}
(If $G^\lambda$ is connected, which is automatic if $G$ is connected
and $\lambda$ is elliptic, then $\Lambda$ is uniquely determined by
$\lambda$; the {\em existence} of $\Lambda$ is an {\em integrality}
constraint on $\lambda$.) Attached to $(\lambda,\Lambda)$ is a
cohomologically induced unitary
representation $\pi(\text{orbit\ }\lambda,\Lambda)$ satisfying
\begin{equation}\label{eq:cohindD}\begin{aligned}
\text{infinitesimal character} &= i\lambda - \rho_L = d\Lambda
-\rho.\\[1ex]
\text{lowest $K$-type} &= \Lambda - 2\rho({\mathfrak u}\cap
{\mathfrak k}) \\
&= i\lambda - \rho({\mathfrak u}\cap
{\mathfrak p}) + \rho({\mathfrak u}\cap {\mathfrak k}).
\end{aligned}
\end{equation}
If $\lambda$ is small, the formula for the lowest $K$-type can fail:
one thing that is true is that this representation of $K$ appears if
the weight is dominant for $K$.
In \cite{VZ}, the representation $\pi(\text{orbit\ }\lambda,\Lambda)$
was called $A_{\mathfrak q}(\Lambda-2\rho({\mathfrak u}))$.
If $G = K$ is compact, then
\begin{equation}\label{eq:cohindcpt}
\pi(\text{orbit\ }\lambda) = \text{repn of highest
weight\ } i\lambda + \rho({\mathfrak
u}).
\end{equation}
If this weight fails to be dominant, then (still in the
compact case) $\pi(\text{orbit\ }\lambda,\Lambda) = 0$. A confusing
but important aspect of this construction is that the same
representation of $G$ may be attached to several different coadjoint
orbits. Still for $G=K$ compact, the trivial representation is attached
to the orbit of $i\rho({\mathfrak u})$ for each of the
($2^{\text{semisimple rank($K$)}}$) different $K$ conjugacy classes of parabolic
subalgebras ${\mathfrak q}$. If we are looking at the trivial
representation inside functions on a homogeneous space $G/H$, then the
requirement \eqref{eq:GmodH} will ``prefer'' only some of these orbits:
different orbits for different $H$.
\end{subequations}
\begin{subequations}\label{se:invts}
{\bf Notational convention.} If $(\pi,V_\pi)$ is a representation of a
group $G$, and $H\subset G$ is a subgroup, we write
\begin{equation}\label{eq:invts}
(\pi^H,V_\pi^H),
\end{equation}
or often just $\pi^H$ for the subspace of $H$-fixed vectors in
$V_\pi$. If $T\in \End(V_\pi)$ preserves $V_\pi^H$, then we will write
\begin{equation}\label{eq:invtops}
\pi^H(T) =_{\text{def}} T|_{V_\pi^H}
\end{equation}
for the restriction of $T$ to the invariant vectors. This notation
may be confusing because we often write a family of representations
of $G$ as something like
\begin{equation}
\{\pi^G_s \mid s\in S\};
\end{equation}
then in the notation $[\pi^G_s]^H$, the superscripts $G$ and $H$ have
entirely different meanings. We hope that no essential ambiguity
arises in this way.
\end{subequations} | 3,528 | 71,259 | en |
train | 0.35.1 | \section{The classical calculation}
\label{sec:R}
\setcounter{equation}{0}
\begin{subequations}\label{se:Rsphere}
Suppose $n\ge 1$ is an integer. Write $O(n)$ for the orthogonal group
of the standard inner product on ${\mathbb R}^n$, and
\begin{equation}
S^{n-1} = \{v\in {\mathbb R}^n \mid \langle v,v\rangle = 1\}
\end{equation}
for the $(n-1)$-dimensional sphere. We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $O(n)$ acts transitively on $S^{n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
O(n)^{e_1} \simeq O(n-1);
\end{equation}
we embed $O(n-1)$ in $O(n)$ by acting on the last $n-1$
coordinates. This shows
\begin{equation}
S^{n-1} \simeq O(n)/O(n-1).
\end{equation}
\end{subequations}
Now Frobenius reciprocity guarantees that if $H\subset G$ are compact
groups, then
\begin{equation}
L^2(G/H) \simeq \sum_{(\pi,V_\pi)\in \widehat G} V_\pi \otimes
(V_\pi^*)^H.
\end{equation}
In words, the multiplicity of an irreducible representation $\pi$ of
$G$ in $L^2(G/H)$ is equal to the dimension of the space of $H$-fixed
vectors in $\pi^*$. So understanding functions on $G/H$ amounts to
understanding representations of $G$ admitting an $H$-fixed
vector. All of the compact homogeneous spaces $G/H$ that we will
consider are {\em Gelfand pairs}, meaning that $\dim (V_{\pi^*})^H \le
1$ for every $\pi \in \widehat G$.
\begin{subequations}\label{se:Rreps}
Here's how that looks for our example. We omit the cases $n=1$ and $n=2$,
which are degenerate versions of the same thing; so assume $n\ge 3$. A
maximal torus in
$O(n)$ is
\begin{equation}
T = SO(2)^{[n/2]},
\end{equation}
so a weight is an $[n/2]$-tuple of integers. For every integer $a
\ge 0$ there is an irreducible representation $\pi^{O(n)}_a$ of highest
weight
\begin{equation}\label{eq:Rdim}
(a,0,\ldots,0), \qquad \dim \pi^{O(n)}_a = \frac{(a+n/2
-1)\prod_{j=1}^{n-3} (a+j)}{(n/2 -1)\cdot (n-3)!}.
\end{equation}
Notice that the polynomial function of $a$ giving the dimension has
degree $n-2$. One natural description of $\pi^{O(n)}_a$ is
\begin{equation}\label{eq:harmonic}
\pi^{O(n)}_a = S^a({\mathbb C}^n)/r^2 S^{a-2}({\mathbb C}^n);
\end{equation}
what we divide by is zero if $a< 2$. We will be interested in the
{\em infinitesimal characters} of the representations $\pi^{O(n)}_a$; that
is, the scalars by which elements of
\begin{equation}
{\mathfrak Z}({\mathfrak o}(n)) =_{\text{def}} U({\mathfrak
o}(n)_{\mathbb C})^{O(n)}
\end{equation}
act on $\pi^{O(n)}_a$. According to Harish-Chandra's theorem,
infinitesimal characters may be identified with Weyl group orbits of
complexified weights. The infinitesimal character of a
finite-dimensional representation of highest weight $\lambda$ is given
by $\lambda+\rho$, with $\rho$ half the sum of the positive
roots. Using the calculation of $\rho$ given in \eqref{eq:Orho}, we get
\begin{equation}\label{eq:Oinfchar}
\text{infinitesimal character}(\pi^{O(n)}_a) = (a+(n-2)/2, (n-4)/2,
(n-6)/2,\cdots).
\end{equation}
The key fact (in the notation explained in \eqref{se:invts}) is that
\begin{equation}\label{eq:Rkey}
\dim [\pi^{O(n)}_a]^{O(n-1)} = 1 \quad (a\ge 0), \qquad \dim\pi^{O(n-1)} = 0
\quad (\pi \not\simeq \pi^{O(n)}_a).
\end{equation}
Therefore
\begin{equation}\label{eq:Osphere}
L^2(S^{n-1}) \simeq \sum_{a=0}^\infty \pi^{O(n)}_a
\end{equation}
as representations of $O(n)$.
If $n=1$, the definition \eqref{eq:harmonic} of $\pi_a^{O(1)}$ is
still reasonable. Then $\pi_a^{O(1)}$ is one-dimensional if $a=0$ or
$1$, and zero for $a\ge 2$. The formula \eqref{eq:Osphere} is still
valid.
If $n=2$, the definition \eqref{eq:harmonic} of $\pi_a^{O(2)}$ is
still reasonable, and \eqref{eq:Osphere} is still valid. Then
$\pi_a^{O(2)}$ is one-dimensional if $a=0$, and two-dimensional for
$a\ge 1$.
\end{subequations}
\begin{subequations}\label{se:Rorbit}
Here is the orbit method perspective. The Lie algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-symmetric matrices; ${\mathfrak
h}_0$ is the subalgebra in which the first row and column are
zero. We can identify ${\mathfrak g}_0^*$ with ${\mathfrak g}_0$ using
the invariant bilinear form
$$B(X,Y) = \tr(XY).$$
Doing that, define
\begin{equation}
a_{\text{orbit}} = a+(n-2)/2
\end{equation}
{\small
\begin{equation}
\lambda(a_{\text{orbit}}) =\begin{pmatrix} 0& a_{\text{orbit}}/2 &
\quad 0&\dots & 0\\ -a_{\text{orbit}}/2 & 0& \quad 0 &\dots &0\\[1ex]
0&0 &\\
\vdots&\vdots & & \text{\Large $0_{(n-2)\times (n-2)}$}\\[-.5ex]
\\ 0& 0 &&&\end{pmatrix} \in ({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
The isotropy group for $\lambda(a_{\text{orbit}})$ is
\begin{equation}
O(n)^{\lambda(a_{\text{orbit}})} = SO(2)\times O(n-2)
=_{\text{def}} L.
\end{equation}
With this notation,
\begin{equation}
\pi_a^{O(n)} = \pi(\text{orbit\ } \lambda(a_{\text{orbit}})).
\end{equation}
The reason this is true is that the infinitesimal character of the
orbit method representation on the right is (by \eqref{eq:cohindD})
\begin{equation}\begin{aligned}
\lambda(a_{\text{orbit}}) -\rho_L &=
(a+(n-2)/2,-(n-4)/2,-(n-6)/2,\cdots)\\
&= \text{infinitesimal character of\ } \pi_a^{O(n)}.\end{aligned}
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a\ge 0$ but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -(n-2)/2.
\end{equation}
For the compact group $O(n)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0, \qquad 0 > a >
-(n-2)/2,
\end{equation}
(for example because the infinitesimal characters of these
representations are singular) so the difference is not important. But
matters will be more interesting in the noncompact case (Section
\ref{sec:Opq}).
\end{subequations}
Back in the general world of a homogeneous space $G/H$ for compact
groups, fix a (positive) $G$-invariant metric on ${\mathfrak g}_0 =
\Lie(G)$, and write
\begin{equation} \Omega_G = -\text{(sum of squares of an orthonormal
basis)}.
\end{equation}
for the corresponding Casimir operator. (We use a minus sign because
natural choices for the metric are negative definite rather than
positive definite.) The $G$-invariant metric on ${\mathfrak g}_0$
defines an $H$-invariant metric on
${\mathfrak g}_0/{\mathfrak h}_0 \simeq T_e(G/H)$, and therefore a
$G$-invariant Riemannian structure on $G/H$. Write
\begin{equation}
L = \text{negative of Laplace-Beltrami operator on $G/H$,}
\end{equation}
a $G$-invariant differential operator. According to
\cite{GGA}*{Exercise II.A4}, the action of $\Omega_G$ on functions on
$G/H$ is equal to the action of $L$. (The Exercise is stated for symmetric
spaces, but the proof on page 568 works in the present setting.) Consequently
\centerline{on an irreducible $G$-representation $\pi \subset
C^\infty(G/H)$,}
\centerline{$L$ acts by the scalar $\pi(\Omega_G)$.}
So we need to be able to calculate these scalars. If $T$ is a maximal
torus in $G$, and $\pi$ has highest weight $\lambda\in {\mathfrak
t}^*$, then
\begin{equation}
\pi(\Omega_G) = \langle \lambda + 2\rho,\lambda \rangle = \langle
\lambda+\rho, \lambda+ \rho\rangle - \langle \rho,\rho\rangle.
\end{equation}
Here $2\rho\in {\mathfrak t}^*$ is the sum of the positive roots. (The
second formula relates this scalar to the infinitesimal character
written in \eqref{eq:Oinfchar} above.)
\begin{subequations}\label{se:Rspec}
Now we're ready to calculate the spectrum of the spherical Laplace
operator $L$. We need to calculate
$\pi^{O(n)}_a(\Omega_{O(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Orho}
2\rho(O(n)) = (n-2, n-4,\cdots,n-2[n/2]).
\end{equation}
(Recall that we have identified weights of $T=SO(2)^{[n/2]}$ with
$[n/2]$-tuples of integers.) Because our highest weight is
\begin{equation}
\lambda = (a,0,\ldots,0),
\end{equation}
we find
\begin{equation}\label{eq:LR}
\pi^{O(n)}_a(\Omega_{O(n)}) = a^2 + (n-2)a = a_{\text{orbit}}^2 - (n-2)^2/4.
\end{equation}
\end{subequations}
\begin{theorem} \label{thm:Rspec} Suppose $n\ge 3$. The eigenvalues of
the (negative)
Laplace-Beltrami operator $L$ on
$S^{n-1}$ are $a^2 + (n-2)a$, for all non-negative integers $a$. The
multiplicity of this eigenvalue is
$$ \frac{(a+n/2 -1)\prod_{j=1}^{n-3} (a+j)}{(n/2 -1)\cdot (n-3)!},$$
a polynomial in $a$ of degree $n-2$.
\end{theorem}
In Sections \ref{sec:C}--\ref{sec:O} we'll repeat this calculation using other
groups. | 3,201 | 71,259 | en |
train | 0.35.2 | \section{The complex calculation}
\label{sec:C}
\setcounter{equation}{0}
\begin{subequations}\label{se:Usphere}
Suppose $n\ge 1$ is an integer. Write $U(n)$ for the unitary group
of the standard Hermitian inner product on ${\mathbb C}^n$, and
\begin{equation}
S^{2n-1} = \{v\in {\mathbb C}^n \mid \langle v,v\rangle = 1\}
\end{equation}
for the $(2n-1)$-dimensional sphere. We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{2n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $U(n)$ acts transitively on $S^{2n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
U(n)^{e_1} \simeq U(n-1);
\end{equation}
we embed $U(n-1)$ in $U(n)$ by acting on the last $n-1$
coordinates. This shows
\begin{equation}
S^{2n-1} \simeq U(n)/U(n-1).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Ureps}
Here is the representation theory. We omit the case $n=1$,
which is a degenerate version of the same thing; so assume $n\ge 2$. A
maximal torus in
$U(n)$ is
\begin{equation}
T = U(1)^n
\end{equation}
so a weight is an $n$-tuple of integers. For all integers $b\ge 0$ and
$c\ge 0$ there is an irreducible representation $\pi^{U(n)}_{b,c}(n)$ of highest
weight
\begin{equation}
(b,0,\ldots,0,-c), \qquad \dim \pi^{U(n)}_{b,c}= \frac{(b+c+n-1)\prod_{j=1}^{n-2}
(b+j)
(c+j)}{(n -1)\cdot [(n-2)!]^2}.
\end{equation}
Notice that the polynomial giving the dimension has degree $2n-3$ in
the variables $b$ and $c$. A
natural description of the representation is
\begin{equation}
\pi^{U(n)}_{b,c} \simeq S^b({\mathbb C}^n) \otimes S^c(\overline{\mathbb
C}^n)/r^2 S^{b-1}({\mathbb C}^n) \otimes S^{c-1}(\overline{\mathbb
C}^n);
\end{equation}
what we divide by is zero if $b$ or $c$ is zero. The space is (a
quotient of) polynomial functions on ${\mathbb C}^n$, homogeneous of
degree $b$ in the holomorphic coordinates and homogeneous of degree
$c$ in the antiholomorphic coordinates.
Using the calculation of $\rho$ given in \eqref{eq:Urho} below, we
find
\begin{small}\begin{equation}\label{eq:Uinflchar}
\hbox{infl.~char.}\left(\pi^{U(n)}_{b,c}\right) = (b+(n-1)/2,(n-3)/2,\cdots,
-(n-3)/2,-(c+(n-1)/2).
\end{equation}\end{small}
The key fact (again in the notation of \eqref{se:invts}) is that
\begin{equation}\label{e:Ckey}
\dim [\pi^{U(n)}_{b,c}]^{U(n-1)} = 1 \quad (b\ge 0, \ c\ge 0),
\qquad \dim\pi^{U(n-1)} = 0 \quad (\pi \not\simeq \pi^{U(n)}_{b,c}).
\end{equation}
Therefore
\begin{equation}
L^2(S^{2n-1}) \simeq \sum_{b\ge 0, \ c\ge 0} \pi^{U(n)}_{b,c}
\end{equation}
as representations of $U(n)$.
We add one more piece of representation-theoretic information,
without explaining yet why it is useful. If we write $U(1)$ for the
multiplication by unit scalars in the first coordinate, then $U(1)$ commutes
with $U(n-1)$. In any representation of $U(n)$, $U(1)$ therefore
preserves the $U(n-1)$-fixed vectors. The last fact is
\begin{equation}\label{e:extraCreps}
\text{$U(1)$ acts on $[\pi^{U(n)}_{b,c}]^{U(n-1)}$ by the weight $b-c$.}
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Corbit}
Here is the orbit method perspective. The Lie algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-hermitian matrices; ${\mathfrak
h}_0$ is the subalgebra in which the last row and column are zero.
We can identify ${\mathfrak
g}_0^*$ with ${\mathfrak g}_0$ using the invariant bilinear form
$$B(X,Y) = \tr(XY).$$
Doing that, define
\begin{equation}
b_{\text{orbit}} = b+(n-1)/2, \qquad c_{\text{orbit}} = c+ (n-1)/2.
\end{equation}
We need also an auxiliary parameter
\begin{equation}
r_{\text{orbit}} = (b_{\text{orbit}}c_{\text{orbit}})^{1/2}.
\end{equation}
Now define a linear functional
{\small
\begin{equation}
\lambda(b_{\text{orbit}},c_{\text{orbit}}) =\begin{pmatrix} i(b_{\text{orbit}} -
c_{\text{orbit}})& r_{\text{orbit}} & \quad 0&\dots & 0\\ -r_{\text{orbit}}
& 0 & \quad 0 &\dots &0\\[1ex] 0&0 &\\ \vdots&\vdots & & \text{\Large
$0_{(n-2)\times (n-2)}$}\\[-.5ex] \\ 0& 0 &&&\end{pmatrix} \in
({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
This skew-hermitian matrix has been constructed to be orthogonal to
${\mathfrak h}_0$, and to have eigenvalues $ib_{\text{orbit}}$,
$-ic_{\text{orbit}}$, and $n-2$ zeros. Its isotropy group is (as long
as $r_{\text{orbit}} \ne 0$)
\begin{equation}
U(n)^{\lambda(b_{\text{orbit}},c_{\text{orbit}})} = U(1)
\times U(n-2) \times U(1) =_{\text{def}} L;
\end{equation}
the first and last $U(1)$ factors are not the usual ``coordinate''
$U(1)$ factors, but rather correspond to the $ib_{\text{orbit}}$ and
$-ic_{\text{orbit}}$ eigenspaces respectively.
With this notation,
\begin{equation}
\pi_{b,c}^{U(n)} = \pi(\text{orbit\ } \lambda(b_{\text{orbit}},c_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $b,c\ge 0$ but rather
\begin{equation}
b_{\text{orbit}} > 0 \iff b > -(n-1)/2, \qquad c_{\text{orbit}} > 0
\iff c > -(n-1)/2.
\end{equation}
For the compact group $U(n)$ we have
\begin{equation}\begin{aligned}
\pi(\text{orbit\ } \lambda(b_{\text{orbit}},c_{\text{orbit}}))=0
\quad &\text{if}\quad 0 > b > -(n-1)/2 \\
&\text{or}\quad 0 > c > -(n-1)/2,\end{aligned}
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:Upq}).
\end{subequations}
\begin{subequations}\label{se:Cspec}
Now we're ready for spectral theory. We need to calculate
$\pi^{U(n)}_{b,c}(\Omega_{U(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Urho}
2\rho(U(n)) = (n-1, n-3,\cdots,-(n-1)).
\end{equation}
(Recall that we have identified weights of $T=U(1)^n$ with
$n$-tuples of integers.) Because our highest weight is
\begin{equation}
\lambda = (b,0,\ldots,-c),
\end{equation}
we find
\begin{equation}\label{eq:LC}\begin{aligned}
\pi^{U(n)}_{b,c}(\Omega_{U(n)}) &= b^2 + c^2 + (n-1)(b+c)\\
&= b_{\text{orbit}}^2 + c_{\text{orbit}}^2 - (n-1)^2/2.
\end{aligned}
\end{equation}
Just as for the representation theory above, we'll add one more piece
of information without explaining why it will be useful:
\begin{equation}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(\Omega_{U(1)}) = (b-c)^2 = b^2 + c^2 - 2bc.
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraCspec}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(2\Omega_{U(n)} - \Omega_{U(1)}) = (b+c)^2 +
(2n-2)(b+c).
\end{equation}
\end{subequations}
\begin{theorem} Suppose $n\ge 2$. The eigenvalues of the (negative)
Laplace-Beltrami operator $L_U$ on
$S^{2n-1}$ are $b^2 + c^2 + (n-1)(b+c)$, for all non-negative integers
$b$ and $c$. The
multiplicity of this eigenvalue is
$$ \frac{(b+c+n-1)\prod_{j=1}^{n-2}
(b+j) \prod_{k=1}^{n-2} (c+k)}{(n -1)\cdot (n-2)! \cdot (n-2)!}$$
a polynomial in $b$ and $c$ of total degree $2n-3$.
A little more precisely, the multiplicity of an eigenvalue $\lambda$
is the sum over all expressions
$$\lambda = b^2 + c^2 + (n-1)(b+c)$$
(with $b$ and $c$ nonnegative integers) of the indicated polynomial in
$b$ and $c$.
\end{theorem}
Let us compute the first few eigenvalues when $n=2$, so that we are
looking at $S^3$. Some numbers are in Table \ref{table:CLaplacian}. We
have also included eigenvalues and multiplicities from the
calculation with $O(4)$ acting on $S^3$, and the peculiar added
calculations from \eqref{e:extraCreps} and \eqref{e:extraCspec}.
\begin{table}\label{table:CLaplacian}
\caption{\bf Casimir eigenvalues and multiplicities on
$S^3$}
\begin{tabular}{|c|c|c|c|c || c | c | c|}
\hline
$b$ & $c$ & \Tstrut\Bstrut\small{$\pi^{U(n)}_{b,c}(\Omega_{U(2)})$}
&{\scriptsize $[\pi^{U(n)}_{b,c}]^{U(1)}(2\Omega_{U(2)} - \Omega_{U(1)})$}
& dim & $a$
& \small{$\pi^{O(4)}_a(\Omega_{O(4)})$} & dim \\[1ex]
\hline
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
\hline
0 & 1 & 2 & 3 & 2 & 1 & 3 & 4\\
1 & 0 & 2 & 3 & 2 &&&\\
\hline
0 & 2 & 6 & 8 & 3 & 2 & 8 & 9\\
1 & 1 & 4 & 8 & 3&&&\\
2 & 0 & 6 & 8 & 3&&&\\
\hline
0 & 3 & 12 & 15 & 4 & 3 & 15 & 16\\
1 & 2 & 8 & 15 & 4 &&&\\
2 & 1 & 8 & 15 & 4 &&&\\
3 & 0 & 12 & 15 & 4 &&&\\
\hline
\end{tabular}
\end{table}
Since each half (left and the right) of the table concerns $S^3$,
there should be some relationship between them. There are indeed
relationships, but they are not nearly as close as one might
expect. What is being calculated in each case is the spectrum of a
Laplace-Beltrami operator. It is rather clear that the spectra are
quite different: the multiplicities calculated with $U(2)$ are
smaller than the multiplicities calculated with $O(4)$, and the actual
eigenvalues are smaller for $U(2)$ as well. | 3,504 | 71,259 | en |
train | 0.35.3 | \begin{subequations}\label{se:Cspec}
Now we're ready for spectral theory. We need to calculate
$\pi^{U(n)}_{b,c}(\Omega_{U(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Urho}
2\rho(U(n)) = (n-1, n-3,\cdots,-(n-1)).
\end{equation}
(Recall that we have identified weights of $T=U(1)^n$ with
$n$-tuples of integers.) Because our highest weight is
\begin{equation}
\lambda = (b,0,\ldots,-c),
\end{equation}
we find
\begin{equation}\label{eq:LC}\begin{aligned}
\pi^{U(n)}_{b,c}(\Omega_{U(n)}) &= b^2 + c^2 + (n-1)(b+c)\\
&= b_{\text{orbit}}^2 + c_{\text{orbit}}^2 - (n-1)^2/2.
\end{aligned}
\end{equation}
Just as for the representation theory above, we'll add one more piece
of information without explaining why it will be useful:
\begin{equation}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(\Omega_{U(1)}) = (b-c)^2 = b^2 + c^2 - 2bc.
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraCspec}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(2\Omega_{U(n)} - \Omega_{U(1)}) = (b+c)^2 +
(2n-2)(b+c).
\end{equation}
\end{subequations}
\begin{theorem} Suppose $n\ge 2$. The eigenvalues of the (negative)
Laplace-Beltrami operator $L_U$ on
$S^{2n-1}$ are $b^2 + c^2 + (n-1)(b+c)$, for all non-negative integers
$b$ and $c$. The
multiplicity of this eigenvalue is
$$ \frac{(b+c+n-1)\prod_{j=1}^{n-2}
(b+j) \prod_{k=1}^{n-2} (c+k)}{(n -1)\cdot (n-2)! \cdot (n-2)!}$$
a polynomial in $b$ and $c$ of total degree $2n-3$.
A little more precisely, the multiplicity of an eigenvalue $\lambda$
is the sum over all expressions
$$\lambda = b^2 + c^2 + (n-1)(b+c)$$
(with $b$ and $c$ nonnegative integers) of the indicated polynomial in
$b$ and $c$.
\end{theorem}
Let us compute the first few eigenvalues when $n=2$, so that we are
looking at $S^3$. Some numbers are in Table \ref{table:CLaplacian}. We
have also included eigenvalues and multiplicities from the
calculation with $O(4)$ acting on $S^3$, and the peculiar added
calculations from \eqref{e:extraCreps} and \eqref{e:extraCspec}.
\begin{table}\label{table:CLaplacian}
\caption{\bf Casimir eigenvalues and multiplicities on
$S^3$}
\begin{tabular}{|c|c|c|c|c || c | c | c|}
\hline
$b$ & $c$ & \Tstrut\Bstrut\small{$\pi^{U(n)}_{b,c}(\Omega_{U(2)})$}
&{\scriptsize $[\pi^{U(n)}_{b,c}]^{U(1)}(2\Omega_{U(2)} - \Omega_{U(1)})$}
& dim & $a$
& \small{$\pi^{O(4)}_a(\Omega_{O(4)})$} & dim \\[1ex]
\hline
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
\hline
0 & 1 & 2 & 3 & 2 & 1 & 3 & 4\\
1 & 0 & 2 & 3 & 2 &&&\\
\hline
0 & 2 & 6 & 8 & 3 & 2 & 8 & 9\\
1 & 1 & 4 & 8 & 3&&&\\
2 & 0 & 6 & 8 & 3&&&\\
\hline
0 & 3 & 12 & 15 & 4 & 3 & 15 & 16\\
1 & 2 & 8 & 15 & 4 &&&\\
2 & 1 & 8 & 15 & 4 &&&\\
3 & 0 & 12 & 15 & 4 &&&\\
\hline
\end{tabular}
\end{table}
Since each half (left and the right) of the table concerns $S^3$,
there should be some relationship between them. There are indeed
relationships, but they are not nearly as close as one might
expect. What is being calculated in each case is the spectrum of a
Laplace-Beltrami operator. It is rather clear that the spectra are
quite different: the multiplicities calculated with $U(2)$ are
smaller than the multiplicities calculated with $O(4)$, and the actual
eigenvalues are smaller for $U(2)$ as well.
The reason for this is that metric $g_{O}$ that we
used in the $O(2n)$ calculation is not the same as the metric $g_{U}$
that we used in the $U(n)$ calculation. There are two aspects to the
difference. Recall that
\begin{equation}\label{eq:Rtan}
T_{e_1}(S^{n-1}) = \{(0,v_2,\cdots,v_{n}) \mid v_j \in {\mathbb
R}\} \simeq {\mathbb R}^{n-1}.
\end{equation}
In this picture, we will see that $g_{O}$ is the usual inner product
on ${\mathbb R}^{n-1}$. In the $U(n)$ picture,
\begin{equation}\label{eq:T2n-1}
T_{e_1}(S^{2n-1}) = \{(it_1,z_2,\cdots,z_{n}) \mid t\in {\mathbb
R},\ z_j \in {\mathbb C}\} \simeq {\mathbb R} + {\mathbb C}^{n-1}.
\end{equation}
In this picture, $g_U$ is actually {\em twice} the usual inner product
on ${\mathbb C}^{n-1}$:
\begin{equation}\label{eq:gOU}
|(0,x_2+iy_2,\cdots,x_n+iy_n)|^2_{g_U} =
2|(0,0,x_2,y_2,\cdots,x_n,y_n)|^2_{g_O}.
\end{equation}
Here is how to see this factor of two. The
Riemannian structure $g_O$ for $O(n)$ is
related to the invariant bilinear form on ${\mathfrak o}(n)$
\begin{equation}\label{eq:Oform}
\langle X,Y\rangle_{O(n)} = (1/2)\tr(XY).
\end{equation}
The reason for the
factor of $1/2$ is so that the form restricts to (minus) the ``standard''
inner product on the Cartan subalgebra ${\mathfrak s}{\mathfrak
o}(2)^{[n/2]} \simeq {\mathbb R}^{[n/2]}$. Now suppose that
$$v\in {\mathbb R}^{n-1} \simeq T_{e_1}(S^{n-1}).$$
The tangent vector $v$ is given by the $n\times n$ skew-symmetric
matrix $A(v)$ with first row $(0,v)$, first column $(0,-v)^t$, and all
other entries zero. Then
\begin{equation}\label{eq:gO}
|v|^2_{g_O} = -\langle A(v),A(v)\rangle_{O(n)} = -(1/2)(\tr(A(v)A(v)))
= |v|^2,
\end{equation}
proving the statement after \eqref{eq:Rtan} about $g_0$.
For similar reasons, $g_U$ is related
to the invariant form on ${\mathfrak u}(n)$
\begin{equation}\label{eq:Uform1}
\langle Z,W\rangle_{U(n)} = \RE\tr(ZW) = (1/2)(\tr(ZW) +
\overline{\tr(ZW)}).
\end{equation}
If $z\in {\mathbb C}^{n-1} \subset T_{e_1}(S^{2n-1})$, then the
tangent vector $z$ is given by the $n\times n$ skew-Hermitian matrix $B(z)$
with first row $(0,z)$, first column $(0,-\overline{z})^t$, and all
other entries zero. Therefore
\begin{equation}\label{eq:gU}
|z|^2_{g_U} = -\langle B(z),B(z)\rangle_{U(n)} = -\RE(\tr(B(z)B(z)) =
2|z|^2.
\end{equation}
Now equations \eqref{eq:gO} and \eqref{eq:gU} prove \eqref{eq:gOU}
Doubling the Riemannian metric has the effect of
dividing the Laplace operator by two, and so dividing the eigenvalues
by two. For this reason, the eigenvalues computed using $U(n)$ ought
to be half of those computed using $O(2n)$.
But that is still not what the table says. The reason is that in the
$U(n)$ picture, there is a ``preferred'' line in each tangent space,
corresponding to the fibration
$$S^1 \rightarrow S^{2n-1} \rightarrow {\mathbb C}{\mathbb P}^{n-1}.$$
In our coordinates in \eqref{eq:T2n-1}, it is the coordinate
$t_1$. The skew-Hermitian matrix $C(it_1)$ involved has $it_1$ in the first
diagonal entry, and all other entries zero.
\begin{equation}
|(it_1,0,\cdots,0)|^2_{g_U} = -\langle C(it_1),C(it_1)\rangle_{U(n)}
= t_1^2 = |(0,t_1,0,0,\cdots,0,0)|^2_{g_O}:
\end{equation}
no factor of two. So the metric attached to the $U(n)$ action is
fundamentally different from the metric attached to the $O(2n)$
action. In the $U(n)$ case, there is a new (non-elliptic) Laplacian $L_{U(1)}$
acting in the direction of the $S^1$ fibration only. The remarks about
metrics above say that
\begin{equation}\label{eq:LOU}
L_O = 2L_U - L_{U(1)}.
\end{equation}
(The reason is that the sum of squares of derivatives in $L_O$ is
almost exactly twice the sum of squares $L_U$; except that this factor
of two is not needed in the direction of the $U(1)$ fibration.)
The ``extra'' calculations \eqref{e:extraCreps} and
\eqref{e:extraCspec} are calculating the spectrum of
$L_{U(1)}$ representation-theoretically; so the column
$$[\pi^{U(n)}_{b,c}]^{U(1)}(2\Omega_{U(2)} - \Omega_{U(1)})$$
in the table above is calculating the spectrum of the classical Laplacian
$L_O$.
Here is a final representation-theoretic statement, explaining how the
$U(n)$ and $O(2n)$ calculations fit together.
\begin{theorem} \label{thm:OUcptbranch} Suppose $n\ge 2$, and $a$ is a
non-negative integer. Using the inclusion $U(n)\subset O(2n)$, we have
$$\pi^{O(2n)}_a|_{U(n)} = \sum_{\substack{0\le b,c \\[.1ex] b+c = a}} \pi^{U(n)}_{b,c}.$$
The contribution of these representations to the spectrum of the
$O(2n)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(2n)}_a(\Omega_{O(2n)}) &= a^2 + (2n-2)a \\
& = (b+c)^2 + 2(n-1)(b+c) \\
&= [\pi^{U(n)}_{b,c}]^{U(n-1)}(2\Omega_{U(n)} - \Omega_{U(1)}).\end{aligned}$$
\end{theorem} | 3,263 | 71,259 | en |
train | 0.35.4 | \section{The quaternionic calculation}
\label{sec:H}
\setcounter{equation}{0}
\begin{subequations}\label{se:Hsphere}
Suppose $n\ge 1$ is an integer. Write $Sp(n)$ for the unitary group
of the standard Hermitian inner product on ${\mathbb H}^n$. This is a
group of ${\mathbb H}$-linear transformations; that is, ${\mathbb
R}$-linear transformations commuting with scalar multiplication by
${\mathbb H}$. Because ${\mathbb H}$ is noncommutative, these scalar
multiplications do {\em not} commute with each other, and so are {\em
not} linear. It is therefore possible and convenient to enlarge
$Sp(n)$ to
\begin{equation}\label{eq:Spbig}
Sp(n) \times Sp(1) = Sp(n)_{\text{linear}} \times
Sp(1)_{\text{scalar}};
\end{equation}
the second factor is scalar multiplication by unit quaternions. This
enlarged group acts on ${\mathbb H}^n$, by the
formula
\begin{equation}
(g_{\text{linear}},z_{\text{scalar}})\cdot v = gvz^{-1};
\end{equation}
we need the inverse to make the right action of scalar multiplication
into a left action. The action preserves length, and so can be
restricted to the $(4n-1)$-dimensional sphere
\begin{equation}
S^{4n-1} = \{v\in {\mathbb H}^n \mid \langle v,v\rangle = 1\}
\end{equation}
We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{4n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $Sp(n)\times Sp(1)$ acts transitively on $S^{2n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
[Sp(n)\times Sp(1)]^{e_1} \simeq Sp(n-1) \times Sp(1)_\Delta.
\end{equation}
Here we embed $Sp(n-1)$ in $Sp(n)$ by acting on the last $n-1$
coordinates, and the last factor is the diagonal subgroup in
$Sp(1)_{\text{linear}}$ (acting on the first coordinate) and
$Sp(1)_{\text{scalar}}$. This shows
\begin{equation}
S^{4n-1} \simeq [Sp(n)\times Sp(1)]/[Sp(n-1) \times Sp(1)_\Delta].
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Hreps}
Here is the representation theory. We omit the case $n=1$,
which is a degenerate version of the same thing; so assume $n\ge 2$. A
maximal torus in
$Sp(n)$ is
\begin{equation}
T = U(1)^n,
\end{equation}
$n$ copies of the unit complex numbers acting diagonally on ${\mathbb
H}^n$. A weight is therefore an $n$-tuple of integers. For all
integers $d\ge e\ge 0$ there is an irreducible representation
\begin{equation}\begin{aligned}
\pi^{Sp(n)}_{d,e} &\text{\ of highest weight\ } (d,e,0,\ldots,0,0), \\[.3ex]
\dim \pi^{Sp(n)}_{d,e}&= \frac{(d+e+2n-1)(d-e+1)\prod_{j=1}^{2n-3}
(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}.
\end{aligned}\end{equation}
A maximal torus in $Sp(1)$ is $U(1)$, and a weight is an integer. For
each integer $f\ge 0$ there is an irreducible representation
\begin{equation}
\pi^{Sp(1)}_f \text{\ of highest weight\ } f,\quad \dim \pi^{Sp(1)}_f =
f+1.
\end{equation}
We are interested in the representations (for $d\ge e \ge 0$)
\begin{small}
\begin{equation}\begin{aligned}
\pi^{Sp(n)\times Sp(1)}_{d,e} &= \pi^{Sp(n)}_{d,e}\otimes
\pi^{Sp(1)}_{d-e} \\
\dim \pi^{Sp(n)\times Sp(1)}_{d,e}&= \frac{(d+e+2n-1)(d-e+1)^2\prod_{j=1}^{2n-3}
(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}.
\end{aligned}\end{equation}\end{small}
Notice that the polynomial giving the dimension has degree $4n-3$.
Using the calculation of $\rho$ given in \eqref{eq:Hrho} below, we
find
\begin{small}\begin{equation}\label{eq:Hinflchar}
\hbox{infl.~char.}(\pi^{Sp(n)\times Sp(1)}_{d,e}) =
(d+n,e+(n-1),n-2,\cdots,1)(d-e+1).
\end{equation}\end{small}
The key fact is that
\begin{equation}\label{e:Hkey}\begin{aligned}
\dim [\pi^{Sp(n)\times Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta} &= 1 \qquad (d\ge e \ge 0),\\
\dim\pi^{Sp(n-1)\times Sp(1)_\Delta} &= 0 \quad (\pi \not\simeq
\pi^{Sp(n)\times Sp(1)}_{d,e}).
\end{aligned}
\end{equation}
Therefore
\begin{equation}
L^2(S^{4n-1}) \simeq \sum_{d\ge e \ge 0} \pi^{Sp(n)\times Sp(1)}_{d,e}
\end{equation}
as representations of $Sp(n)\times Sp(1)$.
Here is one more piece of representation-theoretic information. We saw that
$Sp(n-1)\times Sp(1)_\Delta \subset Sp(n-1)\times Sp(1)\times Sp(1)
\subset Sp(n) \times Sp(1)$; so inside any
representation of $Sp(n)\times Sp(1)$ we get a natural representation of
$Sp(1)\times Sp(1)$ generated by the $Sp(n-1)\times Sp(1)_\Delta$
fixed vectors. The last fact is
\begin{equation}\label{e:extraHreps}\begin{aligned}
\ [Sp(1)\times Sp(1)] &\cdot [\pi^{Sp(n)\times
Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta} \\ &= \text{irr of
highest weight $(d-e,d-e)$.}
\end{aligned}\end{equation}
This representation has infinitesimal character
\begin{equation}\label{eq:Hsubinflchar}\begin{aligned}
\hbox{infl.~char.}\big([Sp(1)\times Sp(1)]&\cdot[\pi^{Sp(n)\times
Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta}\big) \\ &= (d-e+1,d-e+1).
\end{aligned}\end{equation}
\end{subequations}
\begin{subequations}\label{se:Horbit}
Here is the orbit method perspective. (To simplify the notation, we
will discuss only $G=Sp(n)$ rather than $Sp(n) \times Sp(1)$.) The Lie
algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-hermitian quaternionic matrices;
${\mathfrak h}_0$ is the subalgebra in which the last row and column are zero.
Define
\begin{equation}
d_{\text{orbit}} = d+(n-1), \qquad e_{\text{orbit}} = e+ (n-2).
\end{equation}
We need also an auxiliary parameter
\begin{equation}
r_{\text{orbit}} = (d_{\text{orbit}}e_{\text{orbit}})^{1/2}.
\end{equation}
Now define a linear functional
{\small
\begin{equation}
\lambda(d_{\text{orbit}},e_{\text{orbit}}) =\begin{pmatrix} i(d_{\text{orbit}} +
e_{\text{orbit}})& r_{\text{orbit}} & \quad 0&\dots & 0\\ -r_{\text{orbit}}
& 0 & \quad 0 &\dots &0\\[1ex] 0&0 &\\ \vdots&\vdots & & \text{\Large
$0_{(n-2)\times (n-2)}$}\\[-1ex] \\ 0& 0 &&&\end{pmatrix} \in
({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
This skew-hermitian matrix has been constructed to be orthogonal to
${\mathfrak h}_0$, and to be conjugate by $G$ to
\begin{equation}
\begin{pmatrix} id_{\text{orbit}}& 0 &0& \dots & 0\\
0& ie_{\text{orbit}}& 0& \dots & 0\\[1ex] 0& 0 &&\\ \vdots& \vdots&
& \text{\Large $0_{(n-2)\times(n-2)}$}\\
0&0 \end{pmatrix}
\end{equation}
With this notation,
\begin{equation}
\pi_{d,e}^{Sp(n)} = \pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $d\ge e\ge 0$ but rather
\begin{equation}
d_{\text{orbit}} > e_{\text{orbit}} > 0 \iff d + 1 > e > -(n-1).
\end{equation}
For the compact group $Sp(n)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}}))=0
\quad \text{if}\quad 0 > e > -(n-1)
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:Sppq}).
\end{subequations}
\begin{subequations}\label{se:Hspec}
Now we're ready for spectral theory. Because the group is a
product, it is natural to calculate the eigenvalues of the Casimir
operators from the two factors separately. We calculate first
$\pi^{Sp(n)\times Sp(1))}_{d,e}(\Omega_{Sp(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Hrho}
2\rho(Sp(n)) = (2n, 2n-2,\cdots,2).
\end{equation}
Because our highest weight for $Sp(n)$ is
\begin{equation}
\lambda = (d,e,0,\ldots,0),
\end{equation}
we find
\begin{equation}\label{eq:LH}\begin{aligned}
\pi^{Sp(n)}_{d,e}(\Omega_{Sp(n)}) &= d^2 + e^2 + 2nd + 2(n-1)e\\
&=d_{\text{orbit}}^2 + e_{\text{orbit}}^2 - n^2 - (n-1)^2. \end{aligned}
\end{equation}
Similarly
\begin{equation}
\pi^{Sp(n)\times Sp(1)}_{d,e}(\Omega_{Sp(1)}) = (d-e)^2 + 2(d-e) = d^2 + e^2 -
2de + 2(d-e).
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraHspec}\begin{aligned}
\pi^{Sp(n)\times Sp(1)}_{d,e}(2\Omega_{Sp(n)} - \Omega_{Sp(1)}) &= (d+e)^2 +
(4n-2)(d+e)\\ &=\pi^{O(4n)}_{d+e}(\Omega_{O(4n)}).
\end{aligned}\end{equation}
This formula is first of all just an algebraic identity, obtained by
plugging in $a=d+e$ and $4n$ in the formula \eqref{eq:LR}. But it has
a more serious meaning. Let us directly compare the metrics $g_O$ and
$g_{Sp}$ on $S^{4n-1}$, as we did for $g_{U}$ in Section
\ref{sec:C}. We find that on a $(4n-4)$-dimensional subspace of the
tangent space, $g_O$ is some multiple $x\cdot g_{Sp}$; and on the
orthogonal $3$-dimensional subspace (corresponding to the $Sp(1)\simeq
S^3$ fibers of the bundle $S^{4n-1} \rightarrow {\mathbb
P}^{n-1}({\mathbb H})$) there is a different relationship $g_O =
y\cdot g_{Sp}$. (It is not difficult to check by more careful
calculation that $x=2$ and $z=1$, but we are looking here for what is
obvious.) It follows that
$$ L_{O} = xL_{Sp} - zL_{Sp(1)},$$
exactly as in \eqref{eq:LOU}. If now
$$\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_a^{O(4n)},$$
then we conclude (by computing the Laplacian separately in these two
representations) that there is (for all integers $d\ge e \ge 0$) an
algebraic identity
$$ x(d^2+e^2+2nd +2(n-1)e) - z((d-e)^2 + 2(d-e)) = a^2 + (4n-2)a;$$
here $a\ge 0$ is some integer depending on $d$ and $e$. Since every
integer $a \ge 0$ must appear in such an identity, it follows easily
that $x=2$ and $z=1$, and that $a=d+e$. In particular,
\begin{equation}\label{eq:LOSp}
L_{O} = 2L_{Sp} - L_{Sp(1)}.
\end{equation}
This means that the equation \eqref{e:extraHspec} is describing two
calculations of $L_O$, in the subrepresentation
\begin{equation}
\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_{d+e}^{O(4n)}.
\end{equation}
\end{subequations} | 3,806 | 71,259 | en |
train | 0.35.5 | This skew-hermitian matrix has been constructed to be orthogonal to
${\mathfrak h}_0$, and to be conjugate by $G$ to
\begin{equation}
\begin{pmatrix} id_{\text{orbit}}& 0 &0& \dots & 0\\
0& ie_{\text{orbit}}& 0& \dots & 0\\[1ex] 0& 0 &&\\ \vdots& \vdots&
& \text{\Large $0_{(n-2)\times(n-2)}$}\\
0&0 \end{pmatrix}
\end{equation}
With this notation,
\begin{equation}
\pi_{d,e}^{Sp(n)} = \pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $d\ge e\ge 0$ but rather
\begin{equation}
d_{\text{orbit}} > e_{\text{orbit}} > 0 \iff d + 1 > e > -(n-1).
\end{equation}
For the compact group $Sp(n)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}}))=0
\quad \text{if}\quad 0 > e > -(n-1)
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:Sppq}).
\end{subequations}
\begin{subequations}\label{se:Hspec}
Now we're ready for spectral theory. Because the group is a
product, it is natural to calculate the eigenvalues of the Casimir
operators from the two factors separately. We calculate first
$\pi^{Sp(n)\times Sp(1))}_{d,e}(\Omega_{Sp(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Hrho}
2\rho(Sp(n)) = (2n, 2n-2,\cdots,2).
\end{equation}
Because our highest weight for $Sp(n)$ is
\begin{equation}
\lambda = (d,e,0,\ldots,0),
\end{equation}
we find
\begin{equation}\label{eq:LH}\begin{aligned}
\pi^{Sp(n)}_{d,e}(\Omega_{Sp(n)}) &= d^2 + e^2 + 2nd + 2(n-1)e\\
&=d_{\text{orbit}}^2 + e_{\text{orbit}}^2 - n^2 - (n-1)^2. \end{aligned}
\end{equation}
Similarly
\begin{equation}
\pi^{Sp(n)\times Sp(1)}_{d,e}(\Omega_{Sp(1)}) = (d-e)^2 + 2(d-e) = d^2 + e^2 -
2de + 2(d-e).
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraHspec}\begin{aligned}
\pi^{Sp(n)\times Sp(1)}_{d,e}(2\Omega_{Sp(n)} - \Omega_{Sp(1)}) &= (d+e)^2 +
(4n-2)(d+e)\\ &=\pi^{O(4n)}_{d+e}(\Omega_{O(4n)}).
\end{aligned}\end{equation}
This formula is first of all just an algebraic identity, obtained by
plugging in $a=d+e$ and $4n$ in the formula \eqref{eq:LR}. But it has
a more serious meaning. Let us directly compare the metrics $g_O$ and
$g_{Sp}$ on $S^{4n-1}$, as we did for $g_{U}$ in Section
\ref{sec:C}. We find that on a $(4n-4)$-dimensional subspace of the
tangent space, $g_O$ is some multiple $x\cdot g_{Sp}$; and on the
orthogonal $3$-dimensional subspace (corresponding to the $Sp(1)\simeq
S^3$ fibers of the bundle $S^{4n-1} \rightarrow {\mathbb
P}^{n-1}({\mathbb H})$) there is a different relationship $g_O =
y\cdot g_{Sp}$. (It is not difficult to check by more careful
calculation that $x=2$ and $z=1$, but we are looking here for what is
obvious.) It follows that
$$ L_{O} = xL_{Sp} - zL_{Sp(1)},$$
exactly as in \eqref{eq:LOU}. If now
$$\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_a^{O(4n)},$$
then we conclude (by computing the Laplacian separately in these two
representations) that there is (for all integers $d\ge e \ge 0$) an
algebraic identity
$$ x(d^2+e^2+2nd +2(n-1)e) - z((d-e)^2 + 2(d-e)) = a^2 + (4n-2)a;$$
here $a\ge 0$ is some integer depending on $d$ and $e$. Since every
integer $a \ge 0$ must appear in such an identity, it follows easily
that $x=2$ and $z=1$, and that $a=d+e$. In particular,
\begin{equation}\label{eq:LOSp}
L_{O} = 2L_{Sp} - L_{Sp(1)}.
\end{equation}
This means that the equation \eqref{e:extraHspec} is describing two
calculations of $L_O$, in the subrepresentation
\begin{equation}
\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_{d+e}^{O(4n)}.
\end{equation}
\end{subequations}
Here is what we have proven about how the $Sp(n)$ and $O(4n)$
calculations fit together.
\begin{theorem} \label{thm:OSpcptbranch} Suppose $n\ge 2$, and $a$ is
a non-negative integer. Using the map $Sp(n)\times
Sp(1)\rightarrow O(4n)$, we have
$$\pi^{O(4n)}_a|_{Sp(n)\times Sp(1)} = \sum_{\substack{d\ge e \ge 0 \\ d+e= a}}
\pi^{Sp(n)\times Sp(1)}_{d,e}.$$
The contribution of these representations to the spectrum of the
$O(4n)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(4n)}_a(\Omega_{O(4n)}) &= a^2 + (4n-2)a \\
& = (d+e)^2 + (4n-2)(d+e) \\
&= \pi^{Sp(n)\times Sp(1)}_{d,e}(2\Omega_{Sp(n)} - \Omega_{Sp(1)}).\end{aligned}$$
\end{theorem} | 1,792 | 71,259 | en |
train | 0.35.6 | \section{The octonionic calculation}
\label{sec:O}
\setcounter{equation}{0}
We will make no explicit discussion of octonions, except to say that
$F_4$ is related; and that the non-associativity of octonions makes it
impossible to define a ``projective space'' except in octonionic
dimension one. That is why this example is not part of an infinite
family like the real, complex, and quaternionic ones.
\begin{subequations}\label{se:Osphere}
Write $\Spin(9)$ for the compact spin double cover of $SO(9)$. This
group can be defined using a spin representation $\sigma$, which has dimension
$2^{(9-1)/2} = 16$. The representation is real, so we fix a
realization $(\sigma_{\mathbb R},V_{\mathbb R})$ on a
sixteen-dimensional real vector space. Of course the compact group
$\Spin(9)$ preserves a positive definite inner product on $V_{\mathbb
R}$, and
\begin{equation}
S^{15} = \{v\in V_{\mathbb R} \mid \langle v,v\rangle = 1\}
\end{equation}
We choose as a base point
\begin{equation}
v_1 \in S^{15},
\end{equation}
Then $\Spin(9)$ acts transitively on $S^{15}$. (Once one knows that
$F_{4,c}/\Spin(9)$ is a (sixteen-dimensional) rank one Riemannian
symmetric space, and that the action of $\Spin(9)$ on the tangent
space at the base point is the spin representation, then this is
Cartan's result \eqref{e:rank1}.) The isotropy group at $v_1$ is
\begin{equation}
\Spin(9)^{v_1} \simeq \Spin(7)'.
\end{equation}
The embedding of $\Spin(9)^{v_1}$ in $\Spin(9)$ can be described as follows.
First, we write
\begin{equation}
\Spin(8)\subset \Spin(9)
\end{equation}
for the double cover of $SO(8) \subset SO(9)$. Next, we embed
\begin{equation}
\Spin(7)' \ {\buildrel{\text{spin}}\over \longrightarrow}\ \Spin(8).
\end{equation}
(We use the prime to distinguish this subgroup from the double cover
of $SO(7) \subset SO(8)$, which we will call $\Spin(7) \subset
\Spin(8)$.) The way this works is that the spin representation of
$\Spin'(7)$ has dimension $2^{(7-1)/2}= 8$, is real, and preserves a
quadratic form, so $\Spin'(7) \subset SO(8)$. (Another explanation
appears in \eqref{se:bigG2sphere} below.) Now take the double cover
of this inclusion. This shows
\begin{equation}
S^{15} \simeq \Spin(9)/\Spin(7)'.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Oreps}
Here is the representation theory. A maximal torus in $\Spin(9)$ is a
double cover of $SO(2)^4 \subset SO(9)$. A weight is {\em either} a
$4$-tuple of integers (the weights factoring to $SO(2)^4$) {\em or} a
$4$-tuple from ${\mathbb Z} + 1/2$.
For all integers $x\ge 0$ and $y \ge 0$ there is an irreducible representation
\small
\begin{equation}\begin{aligned}
\pi^{\Spin(9)}_{x,y} &\text{\ of highest weight\ } (y/2 +
x,y/2,y/2,y/2),\\[.3ex]
\dim \pi^{\Spin(9)}_{x,y} &=
\frac{(2x+y+7)\prod_{j=1}^3
(x+j)(y+j+1)(y+2j-1)(x+y+j+3)}{7!\cdot 6! \cdot(1/2)}
\end{aligned}\end{equation}
\normalsize
Notice that the polynomial giving the dimension has degree $13$.
Using the calculation of $\rho$ given in \eqref{eq:octrho} below, we
find
\begin{small}
\begin{equation}\label{eq:octinflchar}
\hbox{infl.~char.}(\pi^{\Spin(9)}_{x,y}) =
((2x+y+7)/2,(y+5)/2,(y+3)/2,(y+1)/2).
\end{equation}
\end{small}
The key fact is that
\begin{equation}\label{e:Okey}\begin{aligned}
\dim [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'} &= 1 \qquad (x\ge 0,\ y \ge
0),\\
\dim\pi^{\Spin(7)'} &= 0 \quad (\pi \not\simeq \pi^{\Spin(9)}_{x,y}).
\end{aligned}\end{equation}
Therefore
\begin{equation}
L^2(S^{15}) \simeq \sum_{x\ge 0,\ y \ge 0} \pi^{\Spin(9)}_{x,y}
\end{equation}
as representations of $\Spin(9)$.
Here is one more piece of representation-theoretic information. We saw that
$\Spin(7)'\subset\Spin(8) \subset \Spin(9)$; so inside any
representation of $\Spin(9)$ we get a natural representation of
$\Spin(8)$ generated by the $\Spin(7)'$ fixed vectors. The last fact is
\begin{small}\begin{equation}\label{e:extraOreps}
\Spin(8)\cdot [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'} = \text{irr of
highest weight $(y/2,y/2,y/2,y/2)$.}
\end{equation}\end{small}
This representation has infinitesimal character
\begin{small}\begin{equation}\label{eq:octsubinflchar}
\hbox{infl.~char.}\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}\right) =
((y+6)/2,(y+4)/2,(y+2)/2,y/2).
\end{equation}\end{small}
Here is why this is true. Helgason's theorem about symmetric spaces says that
the representations of $\Spin(8)$ of highest weights
\begin{equation}\label{eq:87'}
(y/2,y/2,y/2,y/2)
\end{equation}
are precisely the ones having a $\Spin(7)'$-fixed
vector, and furthermore this fixed vector is unique. The
corresponding statement for $\Spin(8)/\Spin(7)$ is the case $n=8$ of
Theorem \ref{thm:Rspec}. In that case the highest weights for
$\Spin(8)$ appearing are the multiples of the fundamental weight
$(1,0,0,0)$ (corresponding to the simple root at the end of the
``long'' leg of the Dynkin diagram of $D_4$. For
$\Spin(8)/\Spin(7)'$, the weights appearing must therefore be
multiples of the fundamental weight $(1/2,1/2,1/2,1/2)$ for a simple
root on one of the ``short'' legs of the Dynkin diagram, proving
\eqref{eq:87'}.
To complete the proof of \eqref{e:Okey} using \eqref{eq:87'} we need only the
classical branching theorem for $\Spin(8) \subset \Spin(9)$ (see for example
\cite{KBeyond}*{Theorem 9.16}).
\end{subequations}
\begin{subequations}\label{se:Oorbit}
Here is the orbit method perspective. Define
\begin{equation}
x_{\text{orbit}} = x+ 2, \qquad y_{\text{orbit}} = y+3.
\end{equation}
Then it turns out that there is a $9\times 9$ real skew-symmetric
matrix $\lambda(x_{\text{orbit}}, y_{\text{orbit}})$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) &\in ({\mathfrak
g}_0/{\mathfrak h}_0)^*\\
\lambda(x_{\text{orbit}},y_{\text{orbit}}) &\ \text{has eigenvalues}\\
&\ \text{$\pm i(x_{\text{orbit}}/2+y_{\text{orbit}}/4)$ and
$\pm i(y_{\text{orbit}}/4)$ (three times).} \end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_{x,y}^{\Spin(9)} = \pi(\text{orbit\ }
\lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $x,y \ge 0$ and but rather
\begin{equation}
x_{\text{orbit}},\ y_{\text{orbit}} > 0 \iff x > -2, y > -3.
\end{equation}
For the compact group $\Spin(9)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(x_{\text{orbit}},y_{\text{orbit}}))=0
\quad \text{if}\quad 0 > x > -2 \ \text{or} \ 0 > y > -3;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:spinp9-p}).
\end{subequations}
\begin{subequations}\label{se:Ospec}
Now we're ready for spectral theory. We need to calculate
$\pi^{\Spin(9)}_{x,y}(\Omega_{\Spin(9)})$. The sum of the positive roots is
\begin{equation}\label{eq:octrho}
2\rho(\Spin(9)) = (7,5,3,1).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (y/2 + x,y/2,y/2,y/2),
\end{equation}
we find
\begin{equation}\label{eq:LO}
\pi^{\Spin(9)}_{x,y}(\Omega_{\Spin(9)}) = x^2 + y^2 + xy +8y + 7x.
\end{equation}
Just as for the representation theory above, we'll add one more piece
of information without explaining why it will be useful:
\begin{equation}
\left(\Spin(8) \cdot [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}
\right)(\Omega_{\Spin(8)}) = y^2 + 6y \end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraOspec}\begin{aligned}
\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}\right)(4\Omega_{\Spin(9)} -
3\Omega_{\Spin(8)}) &= (2x+y)^2 + 14(2x+y)\\
&=\pi^{O(16)}_{2x+y}(\Omega_{O(16)}).
\end{aligned}\end{equation}
The last equality can be established exactly as in
\eqref{e:extraHspec}.
\end{subequations}
Here is how the $\Spin(9)$ and $O(16)$ calculations fit together.
\begin{theorem}\label{thm:Ospin9cptbranch} Using the inclusion
$\Spin(9) \subset O(16)$ given by the spin representation, we have
$$\pi_a^{O(16)}|_{\Spin(9)} = \sum_{\substack{x\ge 0,\ y\ge 0
\\[.2ex] 2x+y=a}} \pi_{x,y}^{\Spin(9)}.$$
The contribution of these representations to the spectrum of the
$O(16)$-invariant Laplacian $L_O$ is
$$\begin{aligned}
\pi^{O(16)}_a(\Omega_{O(16)}) &= a^2 + (16-2)a\\ &= (2x+y)^2 +
14(2x+y) \\
&=
\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}]\right)(4\Omega_{\Spin(9)}
- 3\Omega_{\Spin(8)}).\end{aligned}$$
\end{theorem} | 3,266 | 71,259 | en |
train | 0.35.7 | \section{The small $G_2$ calculation}
\label{sec:G2}
\setcounter{equation}{0}
\begin{subequations}\label{se:G2sphere}
Write $G_{2,c}$ for the $14$-dimensional compact connected Lie group
of type $G_2$. There is a $7$-dimensional real representation
$(\tau_{\mathbb R},W_{\mathbb R})$ of
$G_{2,c}$, whose (complexified) weights are zero and the six short
roots. The representation $\tau_{\mathbb R}$ preserves a positive
definite inner product, and so defines inclusions
\begin{equation}
G_{2,c} \hookrightarrow SO(W_{\mathbb R}), \qquad G_{2,c} \hookrightarrow
\Spin(W_{\mathbb R}).
\end{equation}
The corresponding action of $G_{2,c}$ on $S^6$ is transitive. An
isotropy group is isomorphic to $SU(3)$; this is a subgroup generated
by a maximal torus and the long root $SU(2)$s. Therefore
\begin{equation}
S^{6} = \{w\in W_{\mathbb R} \mid \langle v,v\rangle = 1\} \simeq G_{2,c}/SU(3).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:G2reps}
Here is the representation theory. Having identified a subgroup of
$G_{2,c}$ with $SU(3)$, we may as well take for our maximal torus in
$G_{2,c}$ the diagonal torus
\begin{equation}
T = S(U(1)^3) \subset SU(3).
\end{equation}
The weights of $T$ are therefore
\begin{equation}
X^*(T) = \{\lambda=(\lambda_1,\lambda_2,\lambda_3) \mid \lambda_i -
\lambda_j \in {\mathbb Z},\quad \lambda_1+\lambda_2 + \lambda_3 = 0\}.
\end{equation}
For each integer $a\ge 0$ there is an irreducible representation
\begin{equation}\label{eq:G2dim}
\pi_{a} \text{\ highest wt\ }
(2a/3,-a/3,-a/3), \quad
\dim \pi_{a} = \frac{(2a+5)\prod_{j=1}^4 (a+j)}{5!}
\end{equation}
Notice that the polynomial giving the dimension has degree $5$. In
fact it is exactly the polynomial of \eqref{eq:Rdim} giving the dimension of
$\pi_a^{O(7)}$.
Using the calculation of $\rho$ given in \eqref{eq:G2rho} below, we
find
\begin{equation}\label{eq:G2inflchar}
\hbox{infinitesimal character of\ }\pi_a =
((2a+5)/3,-(a+1)/3,-(a+4)/3).
\end{equation}
The key fact is that
\begin{equation}\label{e:G2key}
\dim \pi_{a}^{SU(3)} = 1 \quad (a\ge 0), \qquad
\dim\pi^{SU(3)} = 0 \quad (\pi \not\simeq \pi_{a}).
\end{equation}
Therefore
\begin{equation}
L^2(S^{6}) \simeq \sum_{a\ge 0} \pi_{a}
\end{equation}
as representations of $G_{2,c}$.
\end{subequations}
\begin{subequations}\label{se:G2orbit}
Here is the orbit method perspective. Define
\begin{equation}
a_{\text{orbit}} = a+5/2.
\end{equation}
Then it turns out that there is an element $\lambda(a_{\text{orbit}})
\in {\mathfrak g}_0^*$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(a_{\text{orbit}}) &\in ({\mathfrak g}_0/{\mathfrak h}_0)^*\\
\lambda(a_{\text{orbit}}) &\ \text{is conjugate to}\\
&\quad a_{\text{orbit}}\cdot(2/3,-1/3,-1/3).
\end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_a = \pi(\text{orbit\ }
\lambda(a_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a \ge 0$ and but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -5/2.
\end{equation}
For the compact group $G_{2,c}$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0
\quad \text{if}\quad 0 > a > -5/2;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:G2s}).
\end{subequations}
\begin{subequations}\label{se:G2spec}
Now we're ready for spectral theory. We need to calculate
$\pi_{a}(\Omega_{G_{2,c}})$. The sum of the positive roots is
\begin{equation}\label{eq:G2rho}
2\rho(G_{2,c}) = (10/3,-2/3,-8/3).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (2a/3,-a/3,-a/3)
\end{equation}
we find
\begin{equation}\label{eq:LG2}\begin{aligned}
\pi_a(\Omega_{G_{2,c}}) &= 2a^2/3 + 10a/3 =
2(a^2 +5a)/3\\
&= (2/3)(a_{\text{orbit}}^2 - 25/4) \end{aligned}
\end{equation}
\end{subequations}
Here is how the $G_{2,c}$ and $O(7)$ calculations fit together.
\begin{theorem}
Using the inclusion $G_{2,c}\subset O(7)$, we have
$$\pi^{O(7)}_a|_{G_{2,c}} = \pi^{G_{2,c}}_a.$$
The contribution of these representations to the spectrum of the
$O(7)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(7)}_a(\Omega_{O(7)}) &= a^2 + 5a \\
&= \pi^{G_{2,c}}_a(3\Omega_{G_{2,c}}/2).\end{aligned}$$
\end{theorem}
This is a consequence of the equality of dimensions observed at
\eqref{eq:G2dim}, together with the fact that the inclusion of
$G_{2,c}$ in $O(7)$ carries (some) short roots to (some) short roots. | 1,791 | 71,259 | en |
train | 0.35.8 | \section{The big $G_2$ calculation}
\label{sec:bigG2}
\setcounter{equation}{0}
\begin{subequations} \label{se:bigG2sphere}
Suppose $n$ is an integer at least two. The group $\Spin(2n)$, or
equivalently the Lie algebra ${\mathfrak s}{\mathfrak p}{\mathfrak
i}{\mathfrak n}(2n)$, has an interesting
outer automorphism of order two: conjugation by the orthogonal matrix
\begin{equation}
\sigma = \Ad \begin{pmatrix} 1 & 0 & \cdots & 0 & 0\\ 0 & 1 &
\cdots & 0 & 0 \\ & &
\ddots & & \\ 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & \cdots & 0 & -1
\end{pmatrix}.
\end{equation}
The group of fixed points of $\sigma$ is the ``first $2n-1$ coordinates''
\begin{equation}
\Spin(2n-1) = \Spin(2n)^{\sigma}.
\end{equation}
The automorphism $\sigma$ implements the automorphism of the Dynkin diagram
\setlength{\unitlength}{1cm}
\begin{picture}(2.8,1)(-2.7,.2)
\multiput(-.1,.5)(.7,0){3}{\circle*{.15}}
\multiput(0,.5)(.7,0){2}{\line(1,0){.5}}
\put(1.45,.48){\small{\dots}}
\put(2.0,.5){\circle*{.15}}
\multiput(2.4,-.01)(0,1.03){2}{\circle*{.15}}
\put(2.07,.57){\line(2,3){.25}}
\put(2.07,.43){\line(2,-3){.25}}
\put(2.7,.22){\begin{rotate}{90}$\longleftrightarrow$\end{rotate}}
\end{picture}
\noindent exchanging the two short legs. If $n=4$, the Dynkin diagram
\setlength{\unitlength}{1cm}
\begin{picture}(2.8,1)(-2,.2)
\multiput(1.3,.5)(.5,0){1}{\circle*{.15}}
\put(1.4,.5){\line(1,0){.5}}
\put(2.0,.5){\circle*{.15}}
\multiput(2.4,-.01)(0,1.03){2}{\circle*{.15}}
\put(2.07,.57){\line(2,3){.25}}
\put(2.07,.43){\line(2,-3){.25}}
\end{picture}
\noindent has two additional involutive automorphisms, exchanging the other two
pairs of legs. This gives rise to two additional (nonconjugate)
automorphisms $\sigma'$ and $\sigma''$ of $\Spin(8)$. Their fixed
point groups are isomorphic to $\Spin(7)$, but not conjugate to the
standard one (or to each other). We call them
\begin{equation}
\Spin(8)^{\sigma'} = \Spin(7)', \qquad \Spin(8)^{\sigma''} = \Spin(7)''.
\end{equation}
The full automorphism group of the Dynkin diagram is the symmetric
group $S_3$; $\sigma_0$ and $\sigma_\pm$ are the three
transpositions, any two of which generate $S_3$. The fixed point group
of the full $S_3$ is
\begin{equation}
\Spin(8)^{S_3} = G_{2,c} = \Spin(7) \cap \Spin(7)';
\end{equation}
this is a classical way to construct $G_{2,c}$. It follows that
\begin{equation}
S^7 = \Spin(8)/\Spin(7) \supset \Spin(7)'/G_{2,c}.
\end{equation}
Because the last homogeneous space is also seven-dimensional, the
inclusion is an equality
\begin{equation}
S^7 = \Spin(7)'/G_{2.c}.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:bigG2reps}
Here is the representation theory. We take for our maximal torus in
$\Spin(7)'$ the double cover $T_+$ of
\begin{equation}
SO(2)^3 \subset SO(7).
\end{equation}
The weights of $T_+$ are
\begin{equation}
X^*(T) = \{\lambda=(\lambda_1,\lambda_2,\lambda_3) \mid \lambda_i \in
{\mathbb Z} \text{\ (all $i$) or \ } \lambda_i \in
{\mathbb Z}+1/2 \text{\ (all $i$)}\}.
\end{equation}
For each integer $a\ge 0$ there is an irreducible representation
\begin{equation}\begin{aligned}
\pi^{\Spin(7)'}_{a} &\text{\ highest wt\ }
(a/2,a/2,a/2), \\
\dim \pi^{\Spin(7)'}_{a} &= \frac{(a+3)\prod_{j=1}^5(a+j)}{3\cdot 5!}
\end{aligned}\end{equation}
Notice that the polynomial giving the dimension has degree $6$; in
fact it is exactly the polynomial \eqref{eq:Rdim} giving the dimension of
$\pi_a^{O(8)}$.
Using the calculation of $\rho$ given in \eqref{eq:spinrho} below (or
in \eqref{eq:Orho}) we find
\begin{equation}\label{eq:bigG2inflchar}
\hbox{infl.~char.}(\pi^{\Spin(7)'}_a) =
((a+5)/2,(a+3)/2,(a+1)/2).
\end{equation}
The key fact is that
\begin{equation}\label{e:bigG2key}
\pi_{a}^{O(8)}|_{\Spin(7)'} = \pi_a^{\Spin(7)'}.
\end{equation}
Therefore
\begin{equation}
L^2(S^{7}) \simeq \sum_{a\ge 0} \pi^{\Spin(7)'}_{a}
\end{equation}
as representations of $\Spin(7)'$.
\end{subequations}
\begin{subequations}\label{se:bigG2orbit}
Here is the orbit method perspective. Define
\begin{equation}
a_{\text{orbit}} = a+3.
\end{equation}
Then it turns out that there is a $7\times 7$ skew-symmetric real
matrix $\lambda(a_{\text{orbit}})$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(a_{\text{orbit}}) &\in ({\mathfrak g}_0/{\mathfrak
h}_0)^*\\ \lambda(a_{\text{orbit}}) &\quad\text{has eigenvalues}\ \pm
a_{\text{orbit}}/4 \ \text{(three times)}.
\end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_{a}^{\Spin(7)'} = \pi(\text{orbit\ }
\lambda(a_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a \ge 0$ and but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -3.
\end{equation}
For the compact group $G_{2,c}$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0
\quad \text{if}\quad 0 > a > -3;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:bigncG2}).
\end{subequations}
\begin{subequations}\label{se:bigG2spec}
Now we're ready for spectral theory. We need to calculate
$\pi_{a}(\Omega_{\Spin(7)'})$. The sum of the positive roots is
\begin{equation}\label{eq:spinrho}
2\rho(\Spin(7)') = (5,3,1).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (a/2,a/2,a/2)
\end{equation}
we find
\begin{equation}\label{eq:LbigG2}\begin{aligned}
\pi^{\Spin(7)'}_a(\Omega_{\Spin(7)'}) &= 3a^2/4 + 9a/2 =
3(a^2 +6a)/4\\ &= (3/4)(a_{\text{orbit}}^2 - 9)\\ &=
(3/4)\pi^{O(8)}_a(\Omega_{O(8)}).
\end{aligned}\end{equation}
\end{subequations}
Here is a summary.
\begin{theorem}
Using the inclusion $\Spin(7)'\subset O(8)$, we have
$$\pi^{O(8)}_a|_{\Spin(7)'} = \pi^{\Spin(7)'}_a.$$
The contribution of these representations to the spectrum of the
$O(8)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(8)}_a(\Omega_{O(8)}) &= a^2 + 6a \\
&= \pi^{\Spin(7)'}_a(4\Omega_{\Spin(7)'}/3).\end{aligned}$$
\end{theorem} | 2,495 | 71,259 | en |
train | 0.35.9 | \section{Invariant differential operators}
\label{sec:invt}
\setcounter{equation}{0}
\begin{subequations}\label{se:invt}
Suppose $H\subset G$ is a closed subgroup of a Lie group $G$. Write
\begin{equation}
{\mathbb D}(G/H) = \text{$G$-invariant differential operators on
$G/H$},
\end{equation}
an algebra. Following for example Helgason \cite{GGA}*{pages
274--275}, we wish to understand this algebra and its spectral
theory as a way to understand functions on $G/H$. A first step is to
describe the algebra in terms of the Lie algebras of $G$ and $H$. This
is done in \cite{Hinvt} when $H$ is reductive in $G$ (precisely, when
the Lie algebra ${\mathfrak h}_0$ has an $\Ad(H)$-stable complement in
${\mathfrak g}_0$). Ways to remove this hypothesis have been understood
for a long time; we follow the nice account in \cite{invt}.
Write
\begin{equation}\begin{aligned}
{\mathfrak g}_0 &= \Lie(G) = \text{real left-invariant vector
fields on $G$}\\
{\mathfrak g} &= {\mathfrak g}_0\otimes_{\mathbb R}{\mathbb C} =
\text{complex left-invariant vector fields on $G$}
\end{aligned}\end{equation}
These vector fields act on functions by differentiating ``on the
right:''
\begin{equation}
(Xf)(g) = \frac{d}{dt}\left( f(g\exp(tX))\right)|_{t=0} \qquad (X\in
{\mathfrak g}_0).
\end{equation}
As usual we can therefore identify the enveloping algebra
\begin{equation}
U({\mathfrak g}) = \text{left-invariant complex differential operators
on $G$}.
\end{equation}
We can identify
\begin{equation}
C^\infty(G/H) = \{f\in C^\infty(G) \mid f(xh) = f(x)\quad (x\in G,
h\in H)\}.
\end{equation}
Now consider the space
\begin{equation}
I(G/H) =_{\text{def}} \left[ U({\mathfrak g})\otimes_{U({\mathfrak
h})}{\mathbb C} \right]^{\Ad(H)\otimes 1}
\end{equation}
Before we pass to $\Ad(H)$-invariants, we have only a left
$U({\mathfrak g})$ module: no algebra structure. But
$\Ad(H)$-invariants inherit the algebra structure from $U({\mathfrak
g})\otimes_{\mathbb C} {\mathbb C}$; so $I(G/H)$ is an algebra. The
natural action
\begin{equation}
U({\mathfrak g})\otimes C^\infty(G) \rightarrow C^\infty(G)
\end{equation}
(which is a left algebra action, but comes by differentiating on the
right) restricts to a left algebra action
\begin{equation}
I(G/H) \otimes C^\infty(G/H) \rightarrow C^\infty(G/H)
\end{equation}
on the subspace $C^\infty(G/H) \subset C^\infty(G)$.
Suppose more generally that $(\tau,V_\tau)$ is a finite-dimensional (and
therefore smooth) representation of $H$. Then
\begin{equation}
{\mathcal V}_\tau = G \times_H V_\tau
\end{equation}
is a $G$-equivariant vector bundle on $G/H$. The space of smooth sections is
\begin{equation}
C^\infty({\mathcal V}_\tau) = \{f\in C^\infty(G,V_\tau) \mid f(xh) =
\tau(h)^{-1} f(x) \quad (x\in G, h\in H)\}.
\end{equation}
Now consider the space
\begin{equation}
I^\tau(G/H) = \left[ U({\mathfrak g})\otimes_{U({\mathfrak h})}\End(V_\tau)
\right]^{(\Ad\otimes \Ad)(H)}
\end{equation}
(The group $H$ acts by automorphisms on both the algebra $U({\mathfrak
g})$ and the algebra $\End(V_\tau)$, in the latter case by
conjugation by the operators $\tau(h)$. The $H$-invariants are taken
for the tensor product of these two actions.) Before we pass to
$\Ad(H)$-invariants, we have only a left
$U({\mathfrak g})$ module: no algebra structure. But
$\Ad(H)$-invariants inherit the algebra structure from $U({\mathfrak
g})\otimes_{\mathbb C} \End(V_\tau)$; so $I^\tau(G/H)$ is an algebra. The
natural action
\begin{equation}
[U({\mathfrak g})\otimes_{\mathbb C} \End(V_\tau)] \otimes
C^\infty(G,V_\tau) \rightarrow C^\infty(G,V_\tau)
\end{equation}
(which is a left algebra action, but comes by differentiating on the
right) restricts to a left algebra action
\begin{equation}\label{eq:bundleinvt}
I^\tau(G/H)\otimes C^\infty({\mathcal V}_\tau) \rightarrow
C^\infty({\mathcal V}_\tau).
\end{equation}
\end{subequations}
\begin{proposition}[Helgason \cite{Hinvt}*{Theorem
10}; \cite{HinvtBull}*{pages 758--759}; Koornwinder
\cite{invt}*{Theorem 2.10}]\label{prop:invt} Suppose $H$ is a
closed subgroup of the Lie group $G$. The action \eqref{eq:bundleinvt}
identifies the algebra $I^\tau(G/H)$ with
$${\mathbb D}^\tau(G/H) = \text{$G$-invariant differential operators
on the vector bundle ${\mathcal V}_\tau$}.$$
The action of $I^\tau(G/H)$ on formal power series sections of
${\mathcal V}_\tau$ at the identity is a faithful action.
\end{proposition}
\begin{subequations}\label{se:Helgharm}
Helgason's idea for invariant harmonic analysis (see for example
\cite{GGA}*{Introduction}) is to understand the
spectral theory of the algebra $I(G/H)={\mathbb D}(G/H)$ on
$C^\infty(G/H)$; or, more generally, of ${\mathbb D}^\tau(G/H)$ on
smooth sections of ${\mathcal V}_\tau$. Suppose for example that
${\mathbb D}(G/H)$ is {\em abelian}, and fix an algebra homomorphism
\begin{equation}
\lambda \colon {\mathbb D}(G/H) \rightarrow {\mathbb C}, \qquad
\lambda \in \Max\Spec({\mathbb D}(G/H)).
\end{equation}
Then the collection of simultaneous eigenfunctions
\begin{equation}
C^\infty(G/H)_\lambda =_{\text{def}} \{f\in C^\infty(G/H) \mid Df =
\lambda(D)f \mid D\in {\mathbb D}(G/H)\}
\end{equation}
is naturally a representation of $G$ (by left translation). The
question is for which $\lambda$ the space $C^\infty(G/H)_\lambda$ is
nonzero; and more precisely, what representation of $G$ it carries.
We can define
\begin{equation}
\Spec(G/H) = \{\lambda \in \Max\Spec({\mathbb D}(G/H)) \mid
C^\infty(G/H)_\lambda \ne 0\}.
\end{equation}
All of these remarks apply equally well to vector bundles.
\end{subequations}
\begin{subequations}\label{se:constructinvts1}
How can we identify interesting or computable invariant differential
operators? The easiest way is using the center of the enveloping
algebra
\begin{equation}\label{eq:Zg}
{\mathfrak Z}({\mathfrak g}) =_{\text{def}} U({\mathfrak g})^G.
\end{equation}
(If $G$ is disconnected, this may be a proper subalgebra of the
center.) The obvious map
\begin{equation}\label{eq:Zgi}
i_G\colon {\mathfrak Z}({\mathfrak g}) \rightarrow I^\tau(G/H),
\qquad z \mapsto z\otimes I_{V_\tau}
\end{equation}
is an algebra homomorphism. Here is how the spectral theory of the
differential operators $i_G( {\mathfrak Z}({\mathfrak g}))$ is related
to representation theory. Suppose that $(\pi,E_\pi)$ is a smooth
irreducible representation of $G$. Under a variety of mild assumptions
(for example, if $G$ is reductive and $\pi$ is quasisimple) there is a
homomorphism
\begin{equation}
\chi_\pi\colon {\mathfrak Z}({\mathfrak g}) \rightarrow {\mathbb C}
\end{equation}
called the {\em infinitesimal character of $\pi$} so that
\begin{equation}
d\pi(z) = \chi_\pi(z)\cdot I_{E_\pi}.
\end{equation}
Suppose now that there is a $G$-equivariant inclusion
\begin{equation}\label{eq:harmanalysis}
j_G\colon E_\pi \rightarrow C^\infty(G/H,{\mathcal V}_\tau).
\end{equation}
Finding inclusions like \eqref{eq:harmanalysis} is one of the things
harmonic analysis is about. One reason we care about it is the
consequences for spectral theory:
\begin{equation}\label{eq:spectral1}
i_G(z) \text{\ acts on $j_G(E_\pi) \subset C^\infty({\mathcal V}_\tau)$ by
the scalar $\chi_\pi(z)$} \qquad (z \in {\mathfrak Z}({\mathfrak
g})).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:constructinvts2}
Here is a generalization. Suppose $G_1$ is a subgroup of $G$
normalized by $H$:
\begin{equation}
G_1 \subset G, \qquad \Ad(H)(G_1) \subset G_1.
\end{equation}
(The easiest way for this to happen is for $G_1$ to contain $H$.) Then
$H$ acts on ${\mathfrak Z}({\mathfrak g}_1)$, so we get
\begin{equation}\label{eq:Zg1i}
i_{G_1}\colon {\mathfrak Z}({\mathfrak g}_1)^H \rightarrow I^\tau(G/H),
\qquad z_1 \mapsto z_1\otimes I_{V_\tau}.
\end{equation}
These invariant differential operators are acting along the
submanifolds
\begin{equation}
xG_1/(G_1\cap H) \subset G/H \qquad (x\in G)
\end{equation}
of $G/H$. An example is the first coordinate $G_1=U(1)$ introduced in
\eqref{se:Ureps}, for $H=U(n-1)$. The operator $\Omega_{U(1)}$ on
$S^{2n-1}$ (acting along the fibers of the map $S^{2n-1} \rightarrow
{\mathbb C}{\mathbb P}^{n-1}$) is one of these new invariant
operators. A more interesting example is $G_1=Sp(1)\times Sp(1)$ studied
in \eqref{e:extraHreps}.
Here is how the spectral theory of these new operators is related to
representation theory. The map \eqref{eq:harmanalysis} is (by
Frobenius reciprocity) the same thing as an $H$-equivariant map
\begin{equation}
j_H\colon E_\pi \rightarrow V_\tau
\end{equation}
or equivalently
\begin{equation}
j_H^*\colon V_\tau^* \rightarrow E_\pi^*.
\end{equation}
It makes sense to define
\begin{equation}
(E_\pi^*)^{G_1,j_H} = \text{$G_1$ representation generated by
$j_H^*(V_\tau^*)$} \subset \pi^*.
\end{equation}
If the $G_1$ representation $(\pi^*)^{G_1,j_H}$ has infinitesimal character
$\chi_1^*$ (the contragredient of the infinitesimal character $\chi_1$),
then
\begin{equation} \label{eq:spectral2}
i_{G_1}(z_1) \text{\ acts on $j_G(E_\pi) \subset C^\infty({\mathcal V}_\tau)$ by
the scalar $\chi_1(z_1)$} \qquad (z_1 \in {\mathfrak Z}({\mathfrak
g}_1)^H).
\end{equation}
The homomorphisms $i_G$ of \eqref{eq:Zgi} and \eqref{eq:Zg1i} define an
algebra homomorphism from the abstract (commutative) tensor product
algebra
\begin{equation}\label{eq:Zgprodi}
i_G\otimes i_{G_1} \colon {\mathfrak Z}({\mathfrak g}) \otimes_{\mathbb
C}{\mathfrak Z}({\mathfrak g}_1) \rightarrow I^\tau(G/H).
\end{equation}
The reason for this is that ${\mathfrak Z}({\mathfrak g})$ commutes
with all of $U({\mathfrak g})$.
\end{subequations}
Now that we understand the relationship between representations in
$C^\infty({\mathcal V}_\tau)$ and the spectrum of invariant
differential operators, let us see what the results of Sections
\ref{sec:R}--\ref{sec:bigG2} can tell us: in particular, about the
kernel of the homomorphism $i_G\otimes i_{G_1}$ of \eqref{eq:Zgprodi}.
\begin{subequations}\label{se:Rdiff}
We begin with $G=O(n)$, $H=O(n-1)$ as in Section
\ref{sec:R}. Write $n=2m+\epsilon$, with $\epsilon=0$ or
$1$. A maximal torus in $G$ is
\begin{equation}
T=SO(2)^m,\qquad {\mathfrak t}_0 = {\mathbb
R}^m, \qquad {\mathfrak t} = {\mathbb C}^m.
\end{equation}
The Weyl group $W(O(n))$ acts by permutation and sign changes on
these $m$ coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCO}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(O(n))} =
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^m/W(O(n)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}$ by \eqref{eq:HCO}. According to
\eqref{eq:spectral1} and \eqref{eq:Oinfchar}, the invariant
differential operator $i_G(z)$ will act on $\pi_a\subset
C^\infty(G/H)$ by the scalar
$$ p(a+(n-2)/2, (n-4)/2,\cdots,(n-2m)/2).$$
Recalling that $n-2m=\epsilon=0$ or $1$, we write this as
\begin{equation}\label{eq:Rscalar}
p(a+(n-2)/2, (n-4)/2,\cdots,\epsilon/2).
\end{equation}
\end{subequations} | 3,979 | 71,259 | en |