Some Properties of and Open Problems on Hessian Nilpotent Polynomials
SOME PROPERTIES OF AND OPEN PROBLEMS ON
HESSIAN NILPOTENT POLYNOMIALS
WENHUA ZHAO
Abstract. In the recent work [BE1], [M], [Z1] and [Z2], the well-known
Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN
(Hessian nilpotent) polynomials (the polynomials whose Hessian matrix
are nilpotent) and their (deformed) inversion pairs. In this paper, we
prove several results on HN polynomials, their (deformed) inversion pairs
as well as on the associated symmetric polynomial or formal maps. We
also propose some open problems for further study of these objects.
1. Introduction
In the recent work [BE1], [M], [Z1] and [Z2], the well-known Jacobian
conjecture (see [BCW] and [E]) has been reduced to a problem on HN (Hes-
sian nilpotent) polynomials, i.e. the polynomials whose Hessian matrix are
nilpotent, and their (deformed) inversion pairs. In this paper, we prove some
properties of HN polynomials, the (deformed) inversion pairs of (HN) poly-
nomial, the associated symmetric polynomial or formal maps, the graphs
assigned to homogeneous harmonic polynomials, etc. Another purpose of
this paper is to draw the reader’s attention to some open problems which we
believe will be interesting and important for further study of these objects.
In this section we first discuss some backgrounds and motivations in Sub-
section 1.1 for the study of HN polynomials and their (deformed) inversion
pairs. We also fix some terminology and notation in this subsection that
will be used throughout this paper. Then in Subsection 1.2 we give an
arrangement description of this paper.
Date: November 17, 2021.
2000 Mathematics Subject Classification. 14R15, 32H02, 32A50.
Key words and phrases. Hessian nilpotent polynomials, inversion pairs, harmonic
polynomials, the Jacobian conjecture.
The author has been partially supported by NSA Grant R1-07-0053.
http://arxiv.org/abs/0704.1689v2
2 WENHUA ZHAO
1.1. Background and Motivation. Let z = (z1, z2, . . . , zn) be n free com-
mutative variables. We denote by C[z] (resp.C[[z]]) the algebra of poly-
nomials (resp. formal power series) of z over C. A polynomial or formal
power series P (z) is said to be HN (Hessian nilpotent) if its Hessian matrix
HesP := ( ∂
∂zi∂zj
) are nilpotent. The study of HN polynomials is mainly mo-
tivated by the recent progress achieved in [BE1], [M], [Z1] and [Z2] on the
well-known JC (Jacobian conjecture), which we will briefly explain below.
Recall that the JC first proposed by Keller [Ke] in 1939 claims: for any
polynomial map F of Cn with the Jacobian j(F ) = 1, its formal inverse
map G must also be a polynomial map. Despite intense study for more than
half a century, the conjecture is still open even for the case n = 2. For
more history and known results before 2000 on the Jacobian conjecture, see
[BCW], [E] and references there. In 2003, M. de Bondt, A. van den Essen
([BE1]) and G. Meng ([M]) independently made the following breakthrough
on the JC.
Let Di :=
(1 ≤ i ≤ n) and D = (D1, . . . , Dn). For any P (z) ∈ C[[z]],
denote by∇P (z) the gradient of P (z), i.e. ∇P (z) := (D1P (z), . . . , DnP (z)).
We say a formal map F (z) = z − H(z) is symmetric if H(z) = ∇P (z) for
some P (z) ∈ C[[z]]. Then, the symmetric reduction of the JC achieved in
[BE1] and [M] is that, to prove or disprove the JC, it will be enough to
consider only symmetric polynomial maps. Combining with the classical
homogeneous reduction achieved in [BCW] and [Y], one may further assume
that the symmetric polynomial maps have the form F (z) = z−∇P (z) with
P (z) homogeneous (of degree 4). Note that, in this case the Jacobian condi-
tion j(F ) = 1 is equivalent to the condition that P (z) is HN. For some other
recent results on symmetric polynomial or formal maps, see [BE1]–[BE5],
[EW], [M], [Wr1], [Wr2], [Z1], [Z2] and [EZ].
Based on the homogeneous reduction and the symmetric reduction of
the JC discussed above, the author further showed in [Z2] that the JC is
actually equivalent to the following so-called vanishing conjecture of HN
polynomials.
Conjecture 1.1. (Vanishing Conjecture) Let ∆ :=
i be the
Laplace operator of C[z]. Then, for any HN polynomial P (z) (of homo-
geneous of degree d = 4), ∆mPm+1(z) = 0 when m >> 0.
Furthermore, the following criterion of Hessian nilpotency for formal
power series was also proved in [Z2].
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 3
Proposition 1.2. For any P (z) ∈ C[[z]] with o(P (z)) ≥ 2, the following
statements are equivalent.
(1) P (z) is HN.
(2) ∆mPm = 0 for any m ≥ 1.
(3) ∆mPm = 0 for any 1 ≤ m ≤ n.
One crucial idea of the proofs in [Z2] for the results above is to study
a special formal deformation of symmetric formal maps. More precisely,
let t be a central formal parameter. For any P (z) ∈ C[[z]], we call F (z) =
z−∇P (z) the associated symmetric maps of P (z). Let Ft(z) = z−t∇P (z).
When the order o(P (z)) of P (z) with respect to z is greater than or equal
to 2, Ft(z) is a formal map of C[[t]][[z]] with Ft=1(z) = F (z). Therefore, we
may view Ft(z) as a formal deformation of the formal map F (z). In this
case, one can also show (see [M] or Lemma 3.14 in [Z1]) that the formal
inverse map Gt(z) := F
t (z) of Ft(z) does exist and is also symmetric,
i.e. there exists a unique Qt(z) ∈ C[[t]][[z]] with o(Qt(z)) ≥ 2 such that
Gt(z) = z + t∇Qt(z). We call Qt(z) the deformed inversion pair of P (z).
Note that, whenever Qt=1(z) makes sense, the formal inverse map G(z) of
F (z) is given by G(z) = Gt=1(z) = z + ∇Qt=1(z), so in this case we call
Q(z) := Qt=1(z) the inversion pair of P (z).
Note that, under the condition o(P (z)) ≥ 2, the deformed inversion pair
Qt(z) of P (z) might not be in C[t][[z]], so Qt=1(z) may not make sense.
But, if we assume further that J(Ft)(0) = 1, or equivalently, (HesP )(0) is
nilpotent, then Ft(z) is an automorphism of C[t][[z]], hence so is its inverse
map Gt(z). Therefore, in this case Qt(z) lies in C[t][[z]] and Qt=1(z) makes
sense. Throughout this paper, whenever the inversion pair Q(z) of a poly-
nomial or formal power series P (z) ∈ C[[z]] (not necessarily HN) is under
concern, our assumption on P (z) will always be o(P (z)) ≥ 2 and (HesP )(0)
is nilpotent. Note that, for any HN P (z) ∈ C[[z]] with o(P (z)) ≥ 2, the
condition that (HesP )(0) is nilpotent holds automatically.
For later purpose, let us recall the following formula derived in [Z2] for
the deformed inversion pairs of HN formal power series.
Theorem 1.3. Suppose P (z) ∈ C[[z]] with o(P (z)) ≥ 2 is HN. Then, we
Qt(z) =
2mm!(m+ 1)!
∆mPm+1(z),(1.1)
4 WENHUA ZHAO
From the equivalence of the JC and the VC discussed above, we see
that the study on the HN polynomials and their (deformed) inversion pairs
becomes important and necessary, at least when the JC is concerned. Note
that, due to the identity TrHesP = ∆P , HN polynomials are just a special
family of harmonic polynomials which are among the most classical objects
in mathematics. Even though harmonic polynomials had been very well
studied since the late of the eighteenth century, it seems that not much has
been known on HN polynomials. We believe that these mysterious (HN)
polynomials deserve much more attentions from mathematicians.
1.2. Arrangement. Considering the length of this paper, we here give a
more detailed arrangement description of the paper.
In Section 2, we consider the following two questions. Let P, S, T ∈ C[[z]]
with P = S + T and Q,U, V their inversion pairs, respectively.
Q1: Under what conditions, P is HN iff both S and T are HN?
Q2: Under what conditions, we have Q = U + V ?
We give some sufficient conditions in Theorems 2.1 and 2.7 for the two
questions above. In Section 3, we employ a recursion formula of inversion
pairs derived in [Z1] and Eq. (1.1) above to derive some estimates for the
radius of convergence of inversion pairs of homogeneous (HN) polynomials
(see Propositions 3.1 and 3.3).
For any P (z) ∈ C[[z]], we say it is self-inverting if its inversion pair Q(z)
is P (z) itself. In Section 4, by using a general result on quasi-translations
proved in [B], we derive some properties of HN self-inverting formal power
series P (z). Another purpose of this section is to draw the reader’s attention
to Open Problem 4.8 on classification of HN self-inverting polynomials or
formal power series.
In Section 5, we show in Proposition 5.1, when the base field has char-
acteristic p > 0, the VC, unlike the JC, actually holds for any polynomials
P (z) even without the HN condition on P (z). It also holds in this case for
any HN formal power series. One interesting question (see Open Problem
5.2) is to see if the VC like the JC fails over C when P (z) is allowed to be
any HN formal power series.
In Section 6, we prove a criterion of Hessian nilpotency for homogeneous
polynomials over C (see Theorem 6.1). Considering the criterion in Propo-
sition 1.2, this criterion is somewhat surprising but its proof turns out to
be very simple.
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 5
Section 7 is mainly motivated by the following question raised by M.
Kumar ([K]) and D. Wright ([Wr3]). Namely, for a symmetric formal map
F (z) = z −∇P (z), how to write f(z) := 1
σ2 − P (z) (where σ2 :=
i=1 z
and P (z) itself as formal power series in F (z)? In this section, we derive
some explicit formulas to answer the questions above and also for the same
question for σ2 (see Proposition 7.2). From these formulas, we also show in
Theorem 7.4 that, the VC holds for a HN polynomial P (z) iff one (hence,
all) of σ2, P (z) and f(z) can be written as a polynomial in F , where F (z) =
z −∇P (z) is the associated polynomial maps of P (z).
Finally, in Section 8, we discuss a graph G(P ) assigned to each homo-
geneous harmonic polynomials P (z). The graph G(P ) was first proposed
by the author and later was further studied by Roel Willems in his master
thesis [Wi] under direction of Professor Arno van den Essen. In Subsection
8.1 we give the definition of the graph G(P ) for any homogeneous harmonic
polynomial P (z) and discuss the connectedness reduction (see Corollary 8.5)
which says, to study the VC for homogeneous HN polynomials P (z), it will
be enough to consider the case when the graph G(P ) is connected. In Sub-
section 8.2 we consider a connection of G(P ) with the tree expansion formula
derived in [M] and [Wr2] for the inversion pair Q(z) of P (z) (see also Propo-
sition 8.9). As an application of the connection, we use it to give another
proof for the connectedness reduction discussed in Corollary 8.5.
One final remark on the paper is as follows. Even though we could have
focused only on (HN) polynomials, at least when only the JC is concerned,
we will formulate and prove our results in the more general setting of (HN)
formal power series whenever it is possible.
Acknowledgement: The author is very grateful to Professors Arno van
den Essen, Mohan Kumar and David Wright for inspiring communications
and constant encouragement. Section 7 was mainly motivated by some
questions raised by Professors Mohan Kumar and David Wright. The author
also would like to thank Roel Willems for sending the author his master
thesis in which he has obtained some very interesting results on the graphs
G(P ) of homogeneous harmonic polynomials. At last but not the least, the
author thanks the referee and the editor for many valuable suggestions.
6 WENHUA ZHAO
2. Disjoint Formal Power Series and Their Deformed Inversion
Pairs
Let P, S, T ∈ C[[z]] with P = S + T , and Q, U and V their inversion
pairs, respectively. In this section, we consider the following two questions:
Q1: Under what conditions, P is HN if and only if both S and T are
Q2: Under what conditions, we have Q = U + V ?
We give some answers to the questions Q1 and Q2 in Theorems 2.1 and
2.7, respectively. The results proved here will also be needed in Section 8
when we consider a graph associated to homogeneous harmonic polynomials.
To question Q1 above, we have the following result.
Theorem 2.1. Let S, T ∈ C[[z]] such that 〈∇(DiS),∇(DjT )〉 = 0 for any
1 ≤ i, j ≤ n, where 〈·, ·〉 denotes the standard C-bilinear form of Cn. Let
P = S + T . Then, we have
(a) Hes (S)Hes (T ) = Hes (T )Hes (S) = 0.
(b) P is HN iff both S and T are HN.
Note that statement (b) in the theorem above was first proved by R.
Willems ([Wi]) in a special setting as in Lemma 2.6 below for homogeneous
harmonic polynomials.
Proof: (a) For any 1 ≤ i, j ≤ n, consider the (i, j)th entry of the product
Hes (S)Hes (T ):
∂zi∂zk
∂zk∂zj
= 〈∇(DiS),∇(DjT )〉 = 0.(2.1)
Hence Hes (S) Hes (T ) = 0. Similarly, we have Hes (T ) Hes (S) = 0.
(b) follows directly from (a) and the lemma below. ✷
Lemma 2.2. Let A, B and C be n × n matrices with entries in any com-
mutative ring. Suppose that A = B + C and BC = CB = 0. Then, A is
nilpotent iff both B and C are nilpotent.
Proof: The (⇐) part is trivial because B and C in particular commute
with each other.
To show (⇒), note that BC = CB = 0. So for any m ≥ 1, we have
AmB = (B + C)mB = (Bm + Cm)B = Bm+1.
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 7
Similarly, we have Cm+1 = AmC. Therefore, if AN = 0 for some N ≥ 1, we
have BN+1 = CN+1 = 0. ✷
Note that, for the (⇐) part of (b) in Theorem 2.1, we need only a weaker
condition. Namely, for any 1 ≤ i, j ≤ n,
〈∇(DiS),∇(DjT )〉 = 〈∇(DjS),∇(DiT )〉,
which will ensure that Hes (S) and Hes (T ) commute.
To consider the second question Q2, let us first fix the following notation.
For any P ∈ C[[z]], let A(P ) denote the subalgebra of C[[z]] generated
by all partial derivatives of P (of any order). We also define a sequence
{Q[m](z) |, m ≥ 1} by writing the deformed inversion pair Qt(z) of P (z) as
Qt(z) =
tm−1Q[m](z).(2.2)
Lemma 2.3. For any P ∈ C[[z]], we have
(a) A(P ) is closed under the action of any differential operator of C[z]
with constant coefficients.
(b) For any m ≥ 1, we have Q[m](z) ∈ A(P ).
Proof: (a) Note that, by the definition of A(P ), a formal power series
g(z) ∈ C[[z]] lies in A(P ) iff it can be written (not necessarily uniquely)
as a polynomial in partial derivatives of P (z). Then, by the Leibniz Rule,
it is easy to see that, for any g(z) ∈ A(P ), Dig(z) ∈ A(P ) (1 ≤ i ≤ n).
Repeating this argument, we see that any partial derivative of g(z) is in
A(P ). Hence (a) follows.
(b) Recall that, by Proposition 3.7 in [Z1], we have the following recurrent
formula for Q[m](z) (m ≥ 1) in general:
Q[1](z) = P (z),(2.3)
Q[m](z) =
2(m− 1)
k,l≥1
k+l=m
〈∇Q[k](z),∇Q[l](z)〉.(2.4)
for any m ≥ 2.
By using (a), the recurrent formulas above and induction on m ≥ 1, it is
easy to check that (b) holds too. ✷
Definition 2.4. For any S, T ∈ C[[z]], we say S and T are disjoint to each
other if, for any g1 ∈ A(S) and g2 ∈ A(T ), we have 〈∇g1,∇g2〉 = 0.
8 WENHUA ZHAO
This terminology will be justified in Section 8 when we consider a graph
G(P ) associated to homogeneous harmonic polynomials P .
Lemma 2.5. Let S, T ∈ C[[z]]. Then S and T are disjoint to each other
iff, for any α, β ∈ Nn, we have
〈∇(DαS),∇(DβT )〉 = 0.(2.5)
Proof: The (⇒) part of the lemma is trivial. Conversely, for any g1 ∈
A(S) and g2 ∈ A(T ) (i = 1, 2), we need show
〈∇g1,∇g2〉 = 0.
But this can be easily checked by, first, reducing to the case that g1 and
g2 are monomials of partial derivatives of S and T , respectively, and then
applying the Leibniz rule and Eq. (2.5) above. ✷
A family of examples of disjoint polynomials or formal power series are
given as in the following lemma, which will also be needed later in Section
Lemma 2.6. Let I1 and I2 be two finite subsets of C
n such that, for any
αi ∈ Ii (i = 1, 2), we have 〈α1, α2〉 = 0. Denote by Ai (i = 1, 2) the
completion of the subalgebra of C[[z]] generated by hα(z) := 〈α, z〉 (α ∈ Ii),
i.e. Ai is the set of all formal power series in hα(z) (α ∈ Ii) over C. Then,
for any Pi ∈ Ai (i = 1, 2), P1 and P2 are disjoint.
Proof: First, by a similar argument as the proof for Lemma 2.3, (a), it
is easy to check that Ai (i = 1, 2) are closed under action of any differen-
tial operator with constant coefficients. Secondly, since Ai (i = 1, 2) are
subalgebras of C[[z]], we have A(Pi) ⊂ Ai (i = 1, 2).
Therefore, to show P1 and P2 are disjoint to each other, it will be enough
to show that, for any gi ∈ Ai (i = 1, 2), we have 〈∇g1,∇g2〉 = 0. But this
can be easily checked by first reducing to the case when gi (i = 1, 2) are
monomials of hα(z) (α ∈ Ii), and then applying the Leibniz rule and the
following identity: for any α, β ∈ Cn,
〈∇hα(z),∇hβ(z)〉 = 〈α, β〉.
Now, for the second question Q2 on page 6, we have the following result.
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 9
Theorem 2.7. Let P, S, T ∈ C[[z]] with order greater than or equal to
2, and Qt, Ut, Vt their deformed inversion pairs, respectively. Assume that
P = S + T and S, T are disjoint to each other. Then
(a) Ut and Vt are also disjoint to each other, i.e. for any α, β ∈ N
n, we
∇DαUt(z),∇D
βVt(z)
(b) We further have
Qt = Ut + Vt.(2.6)
Proof: (a) follows directly from Lemma 2.3, (b) and Lemma 2.5.
(b) Let Q[m], U[m] and V[m] (m ≥ 1) be defined as in Eq. (2.2). Hence it
will be enough to show
Q[m] = U[m] + V[m](2.7)
for any m ≥ 1.
We use induction on m ≥ 1. When m = 1, Eq. (2.7) follows from the
condition P = S + T and Eq. (2.3) . For any m ≥ 2, by Eq. (2.4) and the
induction assumption, we have
Q[m] =
2(m− 1)
k,l≥1
k+l=m
〈∇Q[k],∇Q[l]〉
2(m− 1)
k,l≥1
k+l=m
〈∇U[k] +∇V[k],∇U[l] +∇V[l]〉
Noting that, by Lemma 2.3, U[j] ∈ A(S) and V[j] ∈ A(T ) (1 ≤ j ≤ m):
2(m− 1)
k,l≥1
k+l=m
〈∇U[k],∇U[l]〉+
2(m− 1)
k,l≥1
k+l=m
〈∇V[k],∇V[l]〉
Applying the recursion formula Eq. (2.4) to both U[m] and V[m]:
= U[m] + V[m].
As later will be pointed out in Remark 8.11, one can also prove this
theorem by using a tree expansion formula of inversion pairs, which was
derived in [M] and [Wr2], in the setting as in Lemma 2.6.
10 WENHUA ZHAO
From Theorems 2.1, 2.7 and Eqs. (1.1), (2.2), it is easy to see that we
have the following corollary.
Corollary 2.8. Let Pi ∈ C[[z]] (1 ≤ i ≤ k) which are disjoint to each other.
Set P =
i=1 Pi. Then, we have
(a) P is HN iff each Pi is HN.
(b) Suppose that P is HN. Then, for any m ≥ 0, we have
∆mPm+1 =
∆mPm+1i .(2.8)
Consequently, if the VC holds for each Pi, then it also holds for P .
3. Local Convergence of Deformed Inversion Pairs of
Homogeneous (HN) Polynomials
Let P (z) be a formal power series which is convergent near 0 ∈ Cn. Then
the associated symmetric map F (z) = z − ∇P is a well-defined analytic
map from an open neighborhood of 0 ∈ Cn to Cn. If we further assume that
JF (0) = In×n, the formal inverse G(z) = z +∇Q(z) of F (z) is also locally
well-defined analytic map. So the inversion pair Q(z) of P (z) is also locally
convergent near 0 ∈ Cn. In this section, we use the formulas Eqs. (2.4),
(1.1) and the Cauchy estimates to derive some estimates for the radius of
convergence of inversion pairs Q(z) of homogeneous (HN) polynomials P (z)
(see Propositions 3.1 and 3.3).
First let us fix the following notation.
For any a ∈ Cn and r > 0, we denote by B(a, r) (resp.S(a, r)) the open
ball (resp. the sphere) centered at a ∈ C with radius r > 0. The unit
sphere S(0, 1) will also be denoted by S2n−1. Furthermore, we let Ω(a, r)
be the polydisk centered at a ∈ Cn with radius r > 0, i.e. Ω(a, r) := {z ∈
n | |zi − ai| < r, 1 ≤ i ≤ n}. For any subset A ⊂ C
n, we will use Ā to
denote the closure of A in Cn.
For any polynomial P (z) ∈ C[z] and a compact subset D ⊂ Cn, we set
|P |D to be the maximum value of |P (z)| over D. In particular, when D is
the unit sphere S2n−1, we also write |P | = |P |D, i.e.
|P | := max{|P (z)| | z ∈ S2n−1}.(3.1)
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 11
Note that, for any r ≥ 0 and a ∈ B(0, r), we have Ω(a, r) ⊂ B(a, r) ⊂
B(0, 2r). Combining with the well-known Maximum Principle of holomor-
phic functions, we get
Ω(a,r)
≤ |P |
B(a,r)
≤ |P |
B(0,2r)
= |P |S(0,2r).(3.2)
For the inversion pairs Q of homogeneous polynomials P without HN
condition, we have the following estimate for the radius of convergence at
0 ∈ Cn.
Proposition 3.1. Let P (z) be a non-zero homogeneous polynomial (not
necessarily HN) of degree d ≥ 3 and r0 = (n2
d−1|P |)
2−d . Then the inversion
pair Q(z) converges over the open ball B(0, r0).
To prove the proposition, we need the following lemma.
Lemma 3.2. Let P (z) be any polynomial and r > 0. Then, for any a ∈
B(0, r) and m ≥ 1, we have
∣Q[m](a)
nm−1|P |m
S(0,2r)
2m−1r2m−2
.(3.3)
Proof: We use induction on m ≥ 1. First, when m = 1, by Eq. (2.3) we
have Q[1] = P . Then Eq. (3.3) follows from the fact B(a, r) ⊂ B(0, 2r) and
the maximum principle of holomorphic functions.
Assume Eq. (3.3) holds for any 1 ≤ k ≤ m − 1. Then, by the Cauchy
estimates of holomorphic functions (e.g. see Theorem 1.6 in [R]), we have
∣(DiQ[k])(a)
∣Q[k]
Ω(0,r)
nk−1|P |k
B(0,2r)
2k−1r2k−1
.(3.4)
By Eqs. (2.4) and (3.4), we have
|Q[m](a)| ≤
2(m− 1)
k,l≥1
k+l=m
∣〈∇Q[k],∇Q[l]〉
2(m− 1)
k,l≥1
k+l=m
nk−1|P |k
S(0,2r)
2k−1r2k−1
nℓ−1|P |ℓ
S(0,2r)
2ℓ−1r2ℓ−1
nm−1|P |m
S(0,2r)
2m−1r2m−2
12 WENHUA ZHAO
Proof of Proposition 3.1: By Eq. (2.2) , we know that,
Q(z) =
Q[m](z).(3.5)
To show the proposition, it will be enough to show the infinite series
above converges absolutely over B(0, r) for any r < r0.
First, for any m ≥ 1, let Am be the RHS of the inequality Eq. (3.3). Note
that, since P is homogeneous of degree d ≥ 3, we further have
|P |mB(0,2r) =
(2r)d|P |S2n−1
= (2r)dm|P |m.(3.6)
Therefore, for any m ≥ 1, we have
Am = 2
(d−1)m+1nm−1r(d−2)m+2|P |m,(3.7)
and by Lemma 3.2,
|Q[m](a)| ≤ Am(3.8)
for any a ∈ B(0, r).
Since 0 < r < r0 = (n2
d−1|P |)2−d, it is easy to see that
= n2d−1rd−2|P | < 1.
Therefore, by the comparison test, the infinite series in Eq. (3.5) converges
absolutely and uniformly over the open ball B(0, r). ✷
Note that the estimate given in Proposition 3.1 depends on the number n
of variables. Next we show that, with the HN condition on P , an estimate
independent of n can be obtained as follows.
Proposition 3.3. Let P (z) be a homogeneous HN polynomial of degree
d ≥ 4 and set r0 := (2
d+1|P |)
2−d . Then, the inversion pair Q(z) of P (z)
converges over the open ball B(0, r0).
Note that, when d = 2 or 3, by Wang’s Theorem ([Wa]), the JC holds
in general. Hence it also holds for the associated symmetric map F (z) =
z −∇P when P (z) is HN. Therefore Q(z) in this case is also a polynomial
of z and converges over the whole space Cn.
To prove the proposition above, we first need the following two lemmas.
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 13
Lemma 3.4. Let P (z) be a homogeneous polynomial of degree d ≥ 1 and
r > 0. For any a ∈ B(0, r), m ≥ 0 and α ∈ Nn, we have
|(DαPm+1)(a)| ≤
(2r)d(m+1)|P |m+1.(3.9)
Proof: First, by the Cauchy estimates and Eq. (3.2), we have
|(DαPm+1)(a)| ≤
|Pm+1|
Ω(a,r)
|Pm+1|
B(0,2r)
.(3.10)
On the other hand, by the maximum principle and the condition that P
is homogeneous of degree d ≥ 3, we have
|Pm+1|
B(0,2r)
= |P |m+1
B(0,2r)
= |P |m+1
S(0,2r)
= ((2r)d|P |)m+1(3.11)
= (2r)d(m+1)|P |m+1.
Then, combining Eqs. (3.10) and (3.11), we get Eq. (3.9). ✷
Lemma 3.5. For any m ≥ 1, we have
|α|=m
α! ≤ m!
m+ n− 1
(m+ n− 1)!
(n− 1)!
.(3.12)
Proof: First, for any α ∈ Nn with |α| = m, we have α! ≤ m! since the
binomial
is always a positive integer. Therefore, we have
|α|=m
α! ≤ m!
|α|=m
Secondly, note that
|α|=m
1 is just the number of distinct α ∈ Nn with
|α| = m, which is the same as the number of distinct monomials in n free
commutative variables of degree m. Since the latter is well-known to be the
binomial
m+n−1
, we have
|α|=m
α! ≤ m!
m+ n− 1
(m+ n− 1)!
(n− 1)!
Proof of Proposition 3.3: By Eq. (1.1) , we know that,
Q(z) =
∆mPm+1
2mm!(m+ 1)!
.(3.13)
14 WENHUA ZHAO
To show the proposition, it will be enough to show the infinite series
above converges absolutely over B(0, r) for any r < r0.
We first give an upper bound for the general terms in the series Eq. (3.13)
over B(0, r).
Consider
∆mPm+1 = (
D2i )
mPm+1 =
|α|=m
D2αPm+1.(3.14)
Therefore, we have
|∆mPm+1(a)| ≤
|α|=m
|D2αPm+1(a)|
Applying Lemma 3.4 with α replaced by 2α:
|α|=m
(2α)!
(2r)d(m+1)|P |m+1
Noting that (2α)! ≤ [(2α)!!]2 = 22m(α!)2:
|α|=m
22m(α!)2
(2r)d(m+1)|P |m+1
= m!22m+d(m+1)rd(m+1)−2m|P |m+1
|α|=m
Applying Lemma 3.5:
m!(m+ n− 1)!22m+d(m+1)rd(m+1)−2m|P |m+1
(n− 1)!
Therefore, for any m ≥ 1, we have
∆mPm+1
2mm!(m+ 1)!
2m+d(m+1)rd(m+1)−2m|P |m+1(m+ n− 1)!
(m+ 1)!(n− 1)!
.(3.15)
For any m ≥ 1, let Am be the right hand side of Eq. (3.15) above. Then,
by a straightforward calculation, we see that the ratio
2d+1rd−2|P |.(3.16)
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 15
Since r < r0 = (2
d+1|P |)
2−d , it is easy to see that
= 2d+1rd−2|P | < 1.
Therefore, by the comparison test, the infinite series in Eq. (3.13) con-
verges absolutely and uniformly over the open ball B(0, r). ✷
4. Self-Inverting Formal Power Series
Note that, by the definition of inversion pairs (see page 3), Q ∈ C[[z]] is
the inversion pair of P ∈ C[[z]] iff P is the inversion pair of Q. In other
words, the relation that Q and P are inversion pair of each other in some
sense is a duality relation. Naturally, one may ask, for which P (z), it is
self-dual or self-inverting? In this section, we discuss this special family of
polynomials or formal power series.
Another purpose of this section is to draw the reader’s attention to the
problem of classification of (HN) self-inverting polynomials (see Open Prob-
lem 4.8). Even though the classification of HN polynomials seems to be out
of reach at the current time, we believe that the classification of (HN) self-
inverting polynomials is much more approachable.
Definition 4.1. A formal power series P (z) ∈ C[[z]] with o(P (z)) ≥ 2
and (HesP )(0) nilpotent is said to be self-inverting if its inversion pair
Q(z) = P (z).
Following the terminology introduced in [B], we say a formal map F (z) =
z − H(z) with H(z) ∈ C[[z]]×n and o(H(z)) ≥ 1 is a quasi-translation if
j(F )(0) 6= 0 and its formal inverse map is given by G(z) = z +H(z).
Therefore, for any P (z) ∈ C[[z]] with o(P (z)) ≥ 2 and (HesP )(0) nilpo-
tent, it is self-inverting iff the associated symmetric formal map F (z) =
z −∇P (z) is a quasi-translation.
For quasi-translations, the following general result has been proved in
Proposition 1.1 of [B] for polynomial quasi-translations.
Proposition 4.2. A formal map F (z) = z − H(z) with o(H) ≥ 1 and
JH(0) nilpotent is a quasi-translation if and only if JH ·H = 0.
Even though the proposition above was proved in [B] only in the setting
of polynomial maps, the proof given there works equally well for formal
quasi-translations under the condition that JH(0) is nilpotent. Since it has
16 WENHUA ZHAO
also been shown in Proposition 1.1 in [B] that, for any polynomial quasi-
translations F (z) = z −H(z), JH(z) is always nilpotent, so the condition
that JH(0) is nilpotent in the proposition above does not put any extra
restriction for the case of polynomial quasi-translations.
From Proposition 4.2 above, we immediately have the following criterion
for self-inverting formal power series.
Proposition 4.3. For any P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0)
nilpotent, it is self-inverting if and only if 〈∇P,∇P 〉 = 0.
Proof: Since o(P ) ≥ 2 and (HesP )(0) is nilpotent, by Proposition 4.2,
we see that, P (z) ∈ C[[z]] is self-inverting iff J(∇P )·∇P = (HesP )·∇P = 0.
But, on the other hand, it is easy to check that, for any P (z) ∈ C[[z]], we
have the following identity:
(HesP ) · ∇P =
∇〈∇P,∇P 〉.
Therefore, (HesP ) · ∇P = 0 iff ∇〈∇P,∇P 〉 = 0, and iff 〈∇P,∇P 〉 = 0
because o(〈∇P,∇P 〉) ≥ 2. ✷
Corollary 4.4. For any P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpo-
tent, if it is self-inverting, then so is Pm(z) for any m ≥ 1.
Proof: Note that, for any m ≥ 2, we have o(Pm(z)) ≥ 2m > 2 and
(HesP )(0) = 0. Then, the corollary follows immediately from Proposition
4.3 and the following general identity:
〈∇Pm,∇Pm〉 = m2P 2m−2〈∇P,∇P 〉.(4.1)
Corollary 4.5. For any harmonic formal power series P (z) ∈ C[[z]] with
o(P ) ≥ 2 and (HesP )(0) nilpotent, it is self-inverting iff ∆P 2 = 0.
Proof: This follows immediately from Proposition 4.3 and the following
general identity:
∆P 2 = 2(∆P )P + 2〈∇P,∇P 〉.(4.2)
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 17
Proposition 4.6. Let P (z) be a harmonic self-inverting formal power se-
ries. Then, for any m ≥ 1, Pm is HN.
Proof: First, we use the mathematical induction on m ≥ 1 to show that
∆Pm = 0 for any m ≥ 1.
The case of m = 1 is given. For any m ≥ 2, consider
∆Pm = ∆(P · Pm−1)
= (∆P )Pm−1 + P (∆Pm−1) + 2〈∇P,∇Pm−1〉
= (∆P )Pm−1 + P (∆Pm−1) + 2(m− 1)Pm−2〈∇P,∇P 〉.
Then, by the mathematical induction assumption and Proposition 4.3, we
get ∆Pm = 0.
Secondly, for any fixed m ≥ 1 and d ≥ 1, we have
∆d[(Pm)d] = ∆d−1(∆P dm) = 0.
Then, by the criterion in Proposition 1.2, Pm is HN. ✷
Example 4.7. Note that, in Section 5.2 of [Z2], a family of self-inverting
HN formal power series has been constructed as follows.
Let Ξ be any non-empty subset of Cn such that, for any α, β ∈ Ξ, 〈α, β〉 =
0. Let A be the completion of the subalgebra of C[[z]] generated by hα(z) :=
〈α, z〉 (α ∈ Ξ), i.e. A is the set of all formal power series in hα(z) (α ∈ Ξ)
over C. Then it is straightforward to check (or see Section 5.2 of [Z2] for
details) that any element P (z) ∈ A is HN and self-inverting.
It is unknown if all HN self-inverting polynomials or formal power series
can be obtained by the construction above. More generally, we believe the
following open problem is worth investigating.
Open Problem 4.8. (a) Decide whether or not all self-inverting polyno-
mials or formal power series are HN.
(b) Classify all (HN) self-inverting polynomials and formal power series.
Finally, let us point out that, for any self-inverting P (z) ∈ C[[z]], the
deformed inversion pair Qt(z) (not just Q(z) = Qt=1(z)) is also same as
P (z).
Proposition 4.9. Let P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpo-
tent. Then P (z) is self-inverting if and only if Qt(z) = P (z).
18 WENHUA ZHAO
Proof: First, let us point out the following observations.
Let t be a formal central parameter and Ft(z) = z − t∇P (z) as before.
Since o(P ) ≥ 2 and (HesP )(0) is nilpotent, we have j(Ft)(0) = 1. Therefore,
Ft(z) is an automorphism of the algebra C[t][[z]] of formal power series of z
over C[t]. Since the inverse map of Ft(z) is given by Gt(z) = z + t∇Qt(z),
we see that Qt(z) ∈ C[t][[z]]. Therefore, for any t0 ∈ C, Qt=t0(z) makes
sense and lies in C[[z]]. Furthermore, by the uniqueness of inverse maps, it
is easy to see that the inverse map of Ft0 = z− t0∇P of C[t][[z]] is given by
Gt0(z) = z + t0∇Qt=t0 . Therefore the inversion pair of t0P (z) is given by
t0Qt=t0(z).
With the notation and observations above, by choosing t0 = 1, we have
Qt=1(z) = Q(z) and the (⇐) part of the proposition follows immediately.
Conversely, for any t0 ∈ C, we have 〈∇(t0P ),∇(t0P )〉 = t
0〈∇P,∇P 〉. Then,
by Proposition 4.3, t0P (z) is self-inverting and its inversion pair t0Qt=t0(z)
is same as t0P (z), i.e. t0Qt=t0(z) = t0P (z). Therefore, we have Qt=t0(z) =
P (z) for any t0 ∈ C
×. But on the other hand, we have Qt(z) ∈ C[t][[z]]
as pointed above, i.e. the coefficients of all monomials of z in Qt(z) are
polynomials of t, hence we must have Qt(z) = P (z) which is the (⇒) part
of the proposition. ✷
5. The Vanishing Conjecture over Fields of Positive
Characteristic
It is well-known that the JC may fail when F (z) is not a polynomial map
(e.g. F1(z1, z2) = e
−z1; F2(z1, z2) = z2e
z1). It also fails badly over fields of
positive characteristic even in one variable case (e.g. F (x) = x − xp over
a field of characteristic p > 0). However, the situation for the VC over
fields of positive characteristic is dramatically different from the JC even
through these two conjectures are equivalent to each other over fields of
characteristic zero. Actually, as we will show in the proposition below, the
VC over fields of positive characteristic holds for any polynomials (not even
necessarily HN) and also for any HN formal power series.
Proposition 5.1. Let k be a field of characteristic p > 0. Then
(a) For any polynomial P (z) ∈ k[z] (not necessarily homogeneous nor
HN) of degree d ≥ 1, ∆mPm+1 = 0 for any m ≥
d(p−1)
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 19
(b) For any HN formal power series P (z) ∈ k[[z]], i.e. ∆mPm = 0 for
any m ≥ 1, we have, ∆mPm+1 = 0 for any m ≥ p− 1.
In other words, over the fields of positive characteristic, the VC holds
even for HN formal power series P (z) ∈ k[[z]]; while for polynomials, it
holds even without the HN condition nor any other conditions.
Proof: The main reason that the proposition above holds is because of
the following simple fact due to the Leibniz rule and positiveness of the
characteristics of the base field k, namely, for m ≥ 1, u(z), v(z) ∈ k[[z]] and
any differential operator Λ of k[z], we have
Λ(umpv) = umpΛv.(5.1)
Now let P (z) be any polynomial or formal series as in the proposition.
For any m ≥ 1, write m+1 = qmp+rm with qm, rm ∈ Z and 0 ≤ rm ≤ p−1.
Then by Eq. (5.1) , we have
∆mPm+1 = ∆m(P qmpP rm) = P qmp∆mP rm.(5.2)
If P (z) is a polynomial of degree d ≥ 1, we have ∆mP rm = 0 when
d(p−1)
, since in this case 2m > deg(P rm). If P (z) is a HN formal power
series, we have ∆mP rm = 0 when m ≥ p− 1 ≥ rm. Therefore, (a) and (b)
in the proposition follow from Eq. (5.2) and the observations above. ✷
One interesting question is whether or not the VC fails (as the JC does)
for any HN formal power series P (z) ∈ C[[z]] but P (z) 6∈ C[z]? To our best
knowledge, no such counterexample has been known yet. We here put it as
an open problem.
Open Problem 5.2. Find a HN formal power series P (z) ∈ C[[z]] but
P (z) 6∈ C[z], if there are any, such that the VC fails for P (z).
One final remark about Proposition 5.1 is as follows. Note that the crucial
fact used in the proof is that any differential operator Λ of k[z] commutes
with the multiplication operator by the pth power of any element of k[[z]].
Then, by a parallel argument as in the proof of Proposition 5.1, it is easy
to see that the following more general result also holds.
Proposition 5.3. Let k be a field of characteristics p > 0 and Λ a differ-
ential operator of k[z]. Let f ∈ k[[z]]. Assume that, for any 1 ≤ m ≤ p− 1,
there exists Nm > 0 such that Λ
Nmfm = 0. Then, we have Λmfm+1 = 0
when m >> 0.
20 WENHUA ZHAO
In particular, if Λ strictly decreases the degree of polynomials. Then, for
any polynomial f ∈ k[z], we have Λmfm+1 = 0 when m >> 0.
6. A Criterion of Hessian Nilpotency for Homogeneous
Polynomials
Recall that 〈·, ·〉 denotes the standard C bilinear form of Cn. For any
β ∈ Cn, we set hβ(z) := 〈β, z〉 and βD := 〈β,D〉.
The main result of this section is the following criterion of Hessian nilpo-
tency for homogeneous polynomials. Considering the criterion given in
Proposition 1.2, it is somewhat surprising but the proof turns out to be
very simple.
Theorem 6.1. For any β ∈ Cn and homogeneous polynomial P (z) of degree
d ≥ 2, set Pβ(z) := β
D P (z). Then, we have
HesPβ = (d− 2)! (HesP )(β).(6.1)
In particular, P (z) is HN iff, for any β ∈ Cn, Pβ(z) is HN.
To prove the theorem, we need first the following lemma.
Lemma 6.2. Let β ∈ Cn and P (z) ∈ C[z] homogeneous of degree N ≥ 1.
βNDP (z) = N !P (β).(6.2)
Proof: Since both sides of Eq. (6.2) are linear on P (z), we may assume
P (z) is a monomial, say P (z) = za for some a ∈ Nn with |a| = N .
Consider
βNDP (z) = (
βiDi)
Nza =
|k|=N
βkDkza
βaDaza = N !βa = N !P (β).
Proof of Theorem 6.1: We consider
HesPβ(z) =
∂2(βd−2D P )
∂zi∂zj
βd−2D
∂zi∂zj
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 21
Applying Lemma 6.2 to ∂
∂zi∂zj
= (d− 2)!
∂zi∂zj
= (d− 2)! (HesP )(β).
Let {ei | 1 ≤ i ≤ n} be the standard basis of C
n. Applying the theorem
above to β = ei (1 ≤ i ≤ n), we have the following corollary, which was first
proved by M. Kumar [K].
Corollary 6.3. For any homogeneous HN polynomial P (z) ∈ C[z] of degree
d ≥ 2, Dd−2i P (z) (1 ≤ i ≤ n) are also HN.
The reason that we think the criteria given in Theorem 6.1 and Corollary
6.3 interesting is that, Pβ(z) = β
D P (z) is homogeneous of degree 2, and it
is much easier to decide whether a homogeneous polynomial of degree 2 is
HN or not. More precisely, for any homogeneous polynomial U(z) of degree
2, there exists a unique symmetric n× n matrix A such that U(z) = zτAz.
Then it is easy to check that HesU(z) = 2A. Therefore, U(z) is HN iff the
symmetric matrix A is nilpotent.
Finally we end this section with the following open question on the cri-
terion given in Proposition 1.2.
Recall that Proposition 1.2 was proved in [Z2]. We now sketch the argu-
ment.
For any m ≥ 1, we set
um(P ) = TrHes
m(P ),(6.3)
vm(P ) = ∆
mPm.(6.4)
For any k ≥ 1, we define Uk(P ) (resp.Vk(P )) to be the ideal in C[[z]]
generated by {um(P )|1 ≤ m ≤ k} (resp. {vm(P )|1 ≤ m ≤ k}) and all their
partial derivatives of any order. Then it has been shown (in a more general
setting) in Section 4 in [Z2] that Uk(P ) = Vk(P ) for any k ≥ 1.
It is well-known in linear algebra that, if um(P (z)) = 0 when m >> 0,
then HesP is nilpotent and um(P ) = 0 for anym ≥ 1. One natural question
is whether or not this is also the case for the sequence {vm(P ) |m ≥ 1}.
More precisely, we believe the following conjecture which was proposed in
[Z2] is worth investigating.
22 WENHUA ZHAO
Conjecture 6.4. Let P (z) ∈ C[[z]] with o(P (z)) ≥ 2. If ∆mPm(z) = 0 for
m >> 0, then P (z) is HN.
7. Some Results on Symmetric Polynomial Maps
Let P (z) be any formal power series with o(P (z)) ≥ 2 and (HesP )(0)
nilpotent, and F (z) and G(z) as before. Set
σ2 : =
z2i ,(7.1)
f(z) : =
σ2 − P (z).(7.2)
Professors Mohan Kumar [K] and David Wright [Wr3] once asked how to
write P (z) and f(z) in terms of F (z)? More precisely, find U(z), V (z) ∈
C[[z]] such that
U(F (z)) = P (z),(7.3)
V (F (z)) = f(z).(7.4)
In this section, we first derive in Proposition 7.2 some explicit formulas
for U(z) and V (z), and also for W (z) ∈ C[[z]] such that
W (F (z)) = σ2(z).(7.5)
We then show in Theorem 7.4 that, when P (z) is a HN polynomial, the
VC holds for P or equivalently, the JC holds for the associated symmetric
polynomial map F (z) = z −∇P , iff one of U , V and W is polynomial.
Let t be a central parameter and Ft(z) = z− t∇P . Let Gt(z) = z+ t∇Qt
be the formal inverse of Ft(z) as before. We set
ft(z) : =
σ2 − tP (z),(7.6)
Ut(z) : = P (Gt(z)),(7.7)
Vt(z) : = ft(Gt(z)),(7.8)
Wt(z) : = σ2(Gt(z)).(7.9)
Note first that, under the conditions that o(P (z)) ≥ 2 and (HesP )(0) is
nilpotent, we have Gt(z) ∈ C[t][[z]]
×n as mentioned in the proof of Propo-
sition 4.9. Therefore, we have Ut(z), Vt(z),Wt(z) ∈ C[t][[z]], and Ut=1(z),
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 23
Vt=1(z) and Wt=1(z) all make sense. Secondly, from the definitions above,
we have
Wt(z) = 2Vt(z) + 2tUt(z),(7.10)
Ft(z) = ∇ft(z),(7.11)
ft=1(z) = f(z).(7.12)
Lemma 7.1. With the notations above, we have
P (z) = Ut=1(F (z)),(7.13)
f(z) = Vt=1(F (z)),(7.14)
σ2(z) = Wt=1(F (z)).(7.15)
In particular, f(z), P (z) and σ2(z) lie in C[F ] iff Ut=1(z), Vt=1(z) and
Wt=1(z) lie in C[z].
In other words, by setting t = 1, Ut, Vt and Wt will give us U , V and W
in Eqs. (7.3)–(7.5), respectively.
Proof: From the definitions of Ut(z), Vt(z) and Wt(z) (see Eqs. (7.7)–
(7.9), we have
P (z) = Ut(Ft(z)),
ft(z) = Vt(Ft(z)),
σ2(z) = Wt(Ft(z)).
By setting t = 1 in the equations above and noticing that Ft=1(z) = F (z),
we get Eqs. (7.13)–(7.15). ✷
For Ut(z), Vt(z) and Wt(z), we have the following explicit formulas in
terms of the deformed inversion pair Qt of P .
Proposition 7.2. For any formal power series P (z) ∈ C[[z]] (not neces-
sarily HN) with o(P (z)) ≥ 2 and (HesP )(0) nilpotent, we have
Ut(z) = Qt + t
,(7.16)
Vt(z) =
σ2 + t(z
−Qt),(7.17)
Wt(z) = σ2 + 2tz
+ 2t2
.(7.18)
Proof: Note first that, Eq. (7.18) follows directly from Eqs. (7.16), (7.17)
and (7.10).
24 WENHUA ZHAO
To show Eq. (7.16), by Eqs. (3.4) and (3.6) in [Z1], we have
Ut(z) = P (Gt) = Qt +
〈∇Qt,∇Qt〉 = Qt + t
.(7.19)
To show Eq. (7.17), we consider
Vt(z) = ft(Gt)
〈z + t∇Qt(z), z + t∇Qt(z)〉 − tP (Gt)
σ2 + t〈z,∇Qt(z)〉 +
〈∇Qt,∇Qt〉 − tP (Gt)
By Eq. (7.19), substituting Qt +
〈∇Qt,∇Qt〉 for P (Gt):
σ2 + t〈z,∇Qt(z)〉 − tQt(z)
σ2 + t(z
−Qt).
When P (z) is homogeneous and HN, we have the following more explicit
formulas which in particular give solutions to the questions raised by Pro-
fessors Mohan Kumar and David Wright.
Corollary 7.3. For any homogeneous HN polynomial P (z) of degree d ≥ 2,
we have
Ut(z) =
2m(m!)2
∆mPm+1(z)(7.20)
Vt(z) =
(dm − 1)t
2mm!(m+ 1)!
∆mPm+1(z) ,(7.21)
Wt(z) = σ2 +
(dm +m)t
2m−1m!(m+ 1)!
∆mPm+1(z) ,(7.22)
where dm = deg (∆
mPm+1) = d(m+ 1)− 2m (m ≥ 0).
Proof: We give a proof for Eq. (7.20). Eqs. (7.21) can be proved similarly.
(7.22) follows directly from Eqs. Eq. (7.20), (7.21) and (7.10).
By combining Eqs. (7.16) and (1.1), we have
Ut(z) =
tm∆mPm+1(z)
2mm!(m+ 1)!
mtm∆mPm+1(z)
2mm!(m+ 1)!
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 25
= P (z) +
2m(m!)2
∆mPm+1(z)
2m(m!)2
∆mPm+1(z).
Hence, we get Eq. (7.20). ✷
One consequence of the proposition above is the following result on sym-
metric polynomials maps.
Theorem 7.4. For any HN polynomial P (z) (not necessarily homogeneous)
with o(P ) ≥ 2, the following statements are equivalent:
(1) The VC holds for P (z).
(2) P (z) ∈ C[F ].
(3) f(z) ∈ C[F ].
(4) σ2(z) ∈ C[F ].
Note that, the equivalence of the statements (1) and (3) was first proved
by Mohan Kumar ([K]) by a different method.
Proof: Note first that, by Lemma 7.1, it will be enough to show that,
∆mPm+1 = 0 when m >> 0 iff one of Ut(z), Vt(z) and Wt(z) is a polynomial
in t with coefficients in C[z]. Secondly, when P (z) is homogeneous, the
statement above follows directly from Eqs. (7.20)–(7.22).
To show the general case, for any m ≥ 0 and Mt(z) ∈ C[t][[z]], we denote
by [tm](Mt(z)) the coefficient of t
m when we write Mt(z) as a formal power
series of t with coefficients in C[[z]]. Then, from Eqs. (7.16)–(7.18) and
Eq. (1.1), it is straightforward to check that the coefficients of tm (m ≥ 1)
in Ut(z), Vt(z) and Wt(z) are given as follows.
[tm](Ut(z)) =
∆mPm+1
2m(m!)2
,(7.23)
[tm](Vt(z)) =
2m−1(m− 1)!m!
(∆m−1Pm)−∆m−1Pm
,(7.24)
[tm](Wt(z)) =
2m−2(m− 1)!m!
(∆m−1Pm) + (m− 1)∆m−1Pm
(7.25)
26 WENHUA ZHAO
From Eq. (7.23), we immediately have (1) ⇔ (2). To show the equiva-
lences (1) ⇔ (3) and (1) ⇔ (4), note first that o(P ) ≥ 2, so o(∆m−1Pm) ≥ 2
for any m ≥ 1. While, on the other hand, for any polynomial h(z) ∈
C[z] with o(h(z)) ≥ 2, we have, h(z) = 0 iff (z ∂
− 1)h(z) = 0, and iff
+ (m − 1))h(z) = 0 for some m ≥ 1. This is simply because that,
for any monomial zα (α ∈ Nn), we have (z ∂
− 1)zα = (|α| − 1)zα and
+ (m− 1))zα = (|α|+ (m− 1))zα. From this general fact, we see that
(1) ⇔ (3) follows from Eq. (7.24) and (1) ⇔ (4) from Eq. (7.25). ✷
8. A Graph Associated with Homogeneous HN Polynomials
In this section, we would like to draw the reader’s attention to a graph
G(P ) assigned to each homogeneous harmonic polynomials P (z). The graph
G(P ) was first proposed by the author and later was further studied by R.
Willems in his master thesis [Wi] under direction of Professor A. van den
Essen. The introduction of the graph G(P ) is mainly motivated by a crite-
rion of Hessian nilpotency given in [Z2] (see also Theorem 8.2 below), via
which one hopes more necessary or sufficient conditions for a homogeneous
harmonic polynomial P (z) to be HN can be obtained or described in terms
of the graph structure of G(P ).
We first give in Subsection 8.1 the definition of the graph G(P ) for any
homogeneous harmonic polynomial P (z) and discuss the connectedness re-
duction (see Corollary 8.5), i.e. a reduction of the VC to the homogeneous
HN polynomials P such that G(P ) is connected. We then consider in Sub-
section 8.2 a connection of G(P ) with the tree expansion formula derived in
[M] and [Wr2] for the inversion pair Q(z) of P (z) (see Proposition 8.9). As
an application of the connection, we give another proof for the connected-
ness reduction given in Corollary 8.5.
8.1. Definition and the Connectedness Reduction. For any β ∈ Cn,
set hβ(z) := 〈β, z〉 and βD := 〈β,D〉, where 〈·, ·〉 is the standard C-bilinear
form of Cn. Let X(C) denote the set of all isotropic elements of Cn, i.e. the
set of all elements α ∈ Cn such that 〈α, α〉 = 0.
Recall that we have the following fundamental theorem on homogeneous
harmonic polynomials.
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 27
Theorem 8.1. For any homogeneous harmonic polynomial P (z) of degree
d ≥ 2, we have
P (z) =
(z)(8.1)
for some ci ∈ C
× and αi ∈ X(C
n) (1 ≤ i ≤ k).
Note that, replacing αi in Eq. (8.1) by c
i αi, we may also write P (z) as
P (z) =
hdαi(z)(8.2)
with αi ∈ X(C
n) (1 ≤ i ≤ k).
For the proof of Theorem 8.1, see, for example, [I] and [Wi].
We fix a homogeneous harmonic polynomial P (z) ∈ C[z] of degree d ≥ 2,
and assume that P (z) is given by Eq. (8.2) for some αi ∈ X(C
n) (1 ≤ i ≤ k).
We may and will always assume {hdαi(z)|1 ≤ i ≤ k} are linearly independent
in C[z].
Recall the following matrices had been introduced in [Z2]:
AP = (〈αi, αj〉)k×k,(8.3)
ΨP = (〈αi, αj〉h
(z))k×k.(8.4)
Then we have the following criterion of Hessian nilpotency for homoge-
neous harmonic polynomials. For its proof, see Theorem 4.3 in [Z2].
Theorem 8.2. Let P (z) be as above. Then, for any m ≥ 1, we have
TrHes
m(P ) = (d(d− 1))mTrΨmP .(8.5)
In particular, P (z) is HN if and only if the matrix ΨP is nilpotent.
One simple remark on the criterion above is as follows.
Let B be the k × k diagonal matrix with the ith (1 ≤ i ≤ k) diagonal
entry being hαi(z). For any 1 ≤ j ≤ k, set
ΨP ;j := B
d−2−j = (hjαi〈αi, αj〉h
d−2−j
).(8.6)
Then, by repeatedly applying the fact that, for any two k× k matrices C
and D, CD is nilpotent iff so is DC, it is easy to see that Theorem 8.2 can
also be re-stated as follows.
28 WENHUA ZHAO
Corollary 8.3. Let P (z) be given by Eq. (8.2) with d ≥ 2. Then, for any
1 ≤ j ≤ d− 2 and m ≥ 1, we have
TrHes
m(P ) = (d(d− 1))mTrΨmP ;j.(8.7)
In particular, P (z) is HN if and only if the matrix ΨP ;j is nilpotent.
Note that, when d is even, we may choose j = (d− 2)/2. So P is HN iff
the symmetric matrix
ΨP ;(d−2)/2(z) = ( h
(d−2)/2
(z) 〈αi, αj〉 h
(d−2)/2
(z) )(8.8)
is nilpotent.
Motivated by the criterion above, we assign a graph G(P ) to any homo-
geneous harmonic polynomial P (z) as follows.
We fix an expression as in Eq. (8.2) for P (z). The set of vertices of G(P )
will be the set of positive integers [k] := {1, 2, . . . , k}. The vertices i and
j of G(P ) are connected by an edge iff 〈αi, αj〉 6= 0. In this case, we get a
finite graph.
Furthermore, we may also label edges of G(P ) by assigning 〈αi, αj〉 or
(d−2)/2
αi 〈αi, αj〉h
(d−2)/2
αi ), when d is even, for the edge connecting vertices
i, j ∈ [k]. We then get a labeled graph whose adjacency matrix is exactly
AP or ΨP,(d−2)/2 (depending on the labels we choose for the edges of G(P )).
Naturally, one may also ask the following (open) questions.
Open Problem 8.4. (a) Find some necessary or sufficient conditions on
the (labeled) graph G(P ) such that the homogeneous harmonic polynomial
P (z) is HN.
(b) Find some necessary or sufficient conditions on the (labeled) graph
G(P ) such that the VC holds for the homogeneous HN polynomial P (z).
First, let us point out that, to approach the open problems above, it will
be enough to focus on homogeneous harmonic polynomials P such that the
graph G(P ) is connected.
Suppose that the graph G(P ) is a disconnected graph with r ≥ 2 con-
nected components. Let [k] = ⊔ri=1Ii be the corresponding partition of the
set [k] of vertices of G(P ). For each 1 ≤ i ≤ r, we set Pi(z) :=
hdα(z).
Note that, by Lemma 2.6, Pi (1 ≤ i ≤ r) are disjoint to each other, so
Corollary 2.8 applies to the sum P =
i=1 Pi. In particular, we have,
(a) P is HN iff each Pi is HN.
(b) if the VC holds for each Pi, then it also holds for P .
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 29
Therefore, we have the following connectedness reduction.
Corollary 8.5. To study homogeneous HN polynomials P or the VC for
homogeneous HN polynomials P , it will be enough to consider the case when
G(P ) is connected.
Note that, the property (a) above was first proved by R. Willems ([Wi])
by using the criterion in Theorem 8.2. (b) was first proved by the author
by a different argument, and with the author’s permission, it had also been
included in [Wi].
Finally, let us point out that R. Willems ([Wi]) has proved the following
very interesting results on Open Problem 8.4.
Theorem 8.6. ([Wi]) Let P be a homogeneous HN polynomial as in Eq.(8.2)
with d ≥ 4. Let l(P ) be the dimension of the vector subspace of Cn spanned
by {αi | 1 ≤ i ≤ k}. Then
(1) If l(P ) = 1, 2, k−1 or k, the graph G(P ) is totally disconnected (i.e.
G(P ) is the graph with no edges).
(2) If l(P ) = k − 2 and G(P ) is connected, then G(P ) is the complete
bi-graph K(4, k − 4).
(3) In the case of (a) and (b) above, the VC holds.
Furthermore, it has also been shown in [Wi] that, for any homogeneous
HN polynomials P , the graph G(P ) can not be any path nor cycles of any
positive length. For more details, see [Wi].
8.2. Connection with the Tree Expansion Formula of Inversion
Pairs. First let us recall the tree expansion formula derived in [M], [Wr2]
for the inversion pair Q(z).
Let T denote the set of all trees, i.e. the set of all connected and simply
connected finite simple graphs. For each tree T ∈ T, denote by V (T ) and
E(T ) the sets of all vertices and edges of T , respectively. Then we have the
following tree expansion formula for inversion pairs.
Theorem 8.7. ([M], [Wr2]) Let P ∈ C[[z]] with o(P ) ≥ 2 and Q its inver-
sion pair. For any T ∈ T, set
QT,P =
ℓ:E(T )→[n]
v∈V (T )
Dadj(v),ℓP,(8.9)
where adj(v) is the set {e1, e2, . . . , es} of edges of T adjacent to v, and
Dadj(v),ℓ = Dℓ(e1)Dℓ(e2) · · ·Dℓ(es).
30 WENHUA ZHAO
Then the inversion pair Q of P is given by
|Aut(T )|
QT,P .(8.10)
Now we assume P (z) is a homogeneous harmonic polynomial d ≥ 2 and
has expression in Eq. (8.2). Under this assumption, it is easy to see that
QT,P (T ∈ T) becomes
QT,P =
f :V (T )→[k]
ℓ:E(T )→[n]
v∈V (T )
Dadj(v),ℓh
αf(v)
(z).(8.11)
The role played by the graph G(P ) of P is to restrict the maps f : V (T ) →
V (G(P ))(= [k]) in Eq. (8.11) to a special family of maps. To be more precise,
let Ω(T,G(P )) be the set of maps f : V (T ) → [k] such that, for any distinct
adjoint vertices u, v ∈ V (T ), f(u) and f(v) are distinct and adjoint in G(P ).
Then we have the following lemma.
Lemma 8.8. For any f : V (T ) → [k] with f 6∈ Ω(T,G(P )), we have
ℓ:E(T )→[n]
v∈V (T )
Dadj(v),ℓh
αf(v)
(z) = 0.(8.12)
Proof: Let f : V (T ) → [k] as in the lemma. Since f 6∈ Ω(T,G(P )),
there exist distinct adjoint v1, v2 ∈ V (T ) such that, either f(v1) = f(v2) or
f(v1) and f(v2) are not adjoint in the graph G(P ). In any case, we have
〈αf(v1), αf(v2)〉 = 0.
Next we consider contributions to the RHS of Eq. (8.11) from the vertices
v1 and v2. Denote by e the edge of T connecting v1 and v2, and {e1, . . . er}
(resp. {ẽ1, . . . ẽs}) the set of edges connected with v1 (resp. v2) besides the
edge e. Then, for any ℓ : E(T ) → [n], the factor in the RHS of Eq. (8.11)
from the vertices v1 and v2 is the product
Dℓ(e)Dℓ(e1) · · ·Dℓ(er)h
αf(v1)
Dℓ(e)Dℓ(ẽ1) · · ·Dℓ(ẽs)h
αf(v2)
.(8.13)
Define an equivalent relation for maps ℓ : E(T ) → [n] by setting ℓ1 ∼ ℓ2
iff ℓ1, ℓ2 have same image at each edge of T except e. Then, by taking sum
of the terms in Eq. (8.13) over each equivalent class, we get the factor
∇Dℓ(e1) · · ·Dℓ(er)h
αf(v1)
(z), ∇Dℓ(ẽ1) · · ·Dℓ(ẽs)h
αf(v2)
.(8.14)
Note that Dℓ(e1) · · ·Dℓ(er)h
αf(v1)
(z) and Dℓ(ẽ1) · · ·Dℓ(ẽs)h
αf(v2)
(z) are con-
stant multiples of some integral powers of hαf(v1)(z) and hαf(v2)(z), respec-
tively. Therefore, 〈αf(v1), αf(v2)〉(= 0) appears as a multiplicative constant
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 31
factor in the term in Eq. (8.14), which makes the term zero. Hence the
lemma follows. ✷
One immediate consequence of the lemma above is the following propo-
sition.
Proposition 8.9. With the setting and notation as above, we have
QT,P =
f∈Ω(T,G(P ))
ℓ:E(T )→[n]
v∈V (T )
Dadj(v),ℓh
αf(v)
(z).(8.15)
Remark 8.10. (a) For any f ∈ Ω(T,G(P )), {f−1(j) | j ∈ Im(f)} gives a
partition of V (T ) since no two distinct vertices in f−1(j) (j ∈ Im(f)) can
be adjoint. In other words, f is nothing but a proper coloring for the tree T ,
which is also subject to certain more conditions from the graph structure of
G(P ). It is interesting to see that the coloring problem of graphs also plays
a role in the inversion problem of symmetric formal maps.
(b) It will be interesting to see if more results can be derived from the
graph G(P ) via the formulas in Eqs. (8.10) and (8.15).
Remark 8.11. By similar arguments as those in proofs of Lemma 8.8, one
may get another proof for Theorem 2.7 in the setting as in Lemma 2.6.
Finally, as an application of Proposition 8.9 above, we give another proof
for the connectedness reduction given in Corollary 8.5.
Let P as given in Eq. (8.2) with the inversion pair Q. Suppose that
there exists a partition [k] = I1 ⊔ I2 with Ii 6= ∅. Let Pi =
hdα(z)
(i = 1, 2) and Qi the inversion pair of Pi. Then we have P = P1 + P2
and G(P1)⊔G(P2) = G(P ). Therefore, to show the connectedness reduction
discussed in the previous subsection, it will be enough to show Q = Q1+Q2.
But this will follow immediately from Eqs. (8.10), (8.15) and the following
lemma.
Lemma 8.12. Let P , P1 and P2 as above, then, for any tree T ∈ T, we
Ω(T,G(P )) = Ω(T,G(P1)) ⊔ Ω(T,G(P2)).
Proof: For any f ∈ Ω(T,G(P )), f preserves the adjacency of vertices
of G(P ). Since T as a graph is connected, Im(f) ⊂ V (G(P )) as a (full)
subgraph of G(P ) must also be connected. Therefore, Im(f) ⊂ V (G(P1))
32 WENHUA ZHAO
or Im(f) ⊂ V (G(P2)). Hence Ω(T,G(P )) ⊂ Ω(T,G(P1)) ⊔ Ω(T,G(P2)). The
other way of containess is obvious. ✷
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http://arxiv.org/abs/0704.1690
SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 33
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Department of Mathematics, Illinois State University, Normal, IL
61790-4520.
E-mail: wzhao@ilstu.edu.
http://arxiv.org/abs/math-ph/0308035
http://arxiv.org/abs/math/0511214
http://arxiv.org/abs/math/0409534
	1. Introduction
	1.1. Background and Motivation
	1.2. Arrangement
	2. Disjoint Formal Power Series and Their Deformed Inversion Pairs
	3. Local Convergence of Deformed Inversion Pairs of Homogeneous (HN) Polynomials
	4. Self-Inverting Formal Power Series
	5. The Vanishing Conjecture over Fields of Positive Characteristic
	6. A Criterion of Hessian Nilpotency for Homogeneous Polynomials
	7. Some Results on Symmetric Polynomial Maps
	8. A Graph Associated with Homogeneous HN Polynomials
	8.1. Definition and the Connectedness Reduction
	8.2. Connection with the Tree Expansion Formula of Inversion Pairs
	References
