A parachute for the degree of a polynomial in algebraically independent
  ones
A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN
ALGEBRAICALLY INDEPENDENT ONES
S. VÉNÉREAU
Abstract. We give a simpler proof as well as a generalization of the main
result of an article of Shestakov and Umirbaev ([3]). This latter article being
the first of two that solve a long-standing conjecture about the non-tameness,
or ”wildness”, of Nagata’s automorphism. As corollaries we get interesting
informations about the leading terms of polynomials forming an automorphism
of K[x1, · · · , xn] and reprove the tameness of automorphisms of K[x1, x2].
The following notations are fixed throughout the article: K is a field of cha-
racteristic 0 and K[x1, · · · , xn] is the ring of polynomials in the n indetermi-
nates x1, · · · , xn with coefficients in K, endowed with the classical degree func-
tion: deg. We consider m algebraically independent polynomials in K[x1, · · · , xn]:
f1, · · · , fm of respective degrees d1, · · · , dm. There is also, for every polynomial G ∈
K[f1, · · · , fm] a unique one G(X1, · · · , Xm) ∈ K[X1, · · · , Xm], where X1, · · · , Xm
are new indeterminates, such that G = G(f1, · · · , fm). By abuse of notation we will
write ∂G
to denote ∂G
(f1, · · · , fm), ∀1 ≤ i ≤ m and degfi G to denote degXi G,
the degree of G in Xi.
The following definition and properties are only formally new, and come from
Definition. We call the parachute of f1, · · · , fm and denote ∇ = ∇(f1, · · · , fm)
the integer
∇ = d1 + · · ·+ dm −m− max
1≤i1,··· ,im≤n
deg jxi1 ,··· ,xim (f1, · · · , fm)
where jxi1 ,··· ,xim (f1, · · · , fm) is the jacobian determinant of f1, · · · , fm with respect
to xi1 , · · · , xim that is jxi1 ,··· ,xim (f1, · · · , fm) = det(∂fi/∂xij )i,j.
Properties. The parachute of f1, · · · , fm has the following estimate:
0 ≤ ∇ = ∇(f1, · · · , fm) ≤ d1 + · · ·+ dm −m.(1)
For any G ∈ K[f1, · · · , fm] and ∀1 ≤ i ≤ m, one has
degG ≥ deg ∂G
+ di −∇ and, inductively,
degG ≥ deg ∂
+ kdi − k∇, ∀k ≥ 0.
Proof. The left minoration 0 ≤ ∇ in (1) is an easy exercise. The right majoration
is a direct consequence of the following
Fact. The vectors gradf1 = (∂f1/∂x1, · · · , ∂f1/∂xn), · · · , gradfm = (∂fm/∂x1, · · · , ∂fm/∂xn)
are linearly independent over K[x1, · · · , xn] therefore the minors of order m of the
matrix
gradf1
gradfm
 are not all 0 and the number max1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm)
is non-negative.
Proof. As mentioned in [3], it is a well-known fact (see e.g. [1] or [2] for a nice proof)
that n rational functions f1, · · · , fn ∈ K(x1, · · · , xn) are algebraically independant
http://arxiv.org/abs/0704.1561v2
2 S. VÉNÉREAU
if and only if their jacobian determinant is not zero. Our fact is then proved
by completing our m algebraically independant polynomials to get n algebraically
independant rational functions: the jacobian determinant is not zero and it follows
that gradf1, · · · , gradfm must be linearly independant. �
It is clearly sufficient to show (2) for i = m.
Take any m integers 1 ≤ i1, · · · , im ≤ n. From the definition of jxi1 ,··· ,xim it is
clear that
deg jxi1 ,··· ,xim (f1, · · · , fm−1, G) ≤ d1 − 1 + · · ·+ dm−1 − 1 + degG− 1
≤ d1 + · · ·+ dm−1 −m+ degG .
On the other hand the chain rule gives
jxi1 ,··· ,xim (f1, · · · , fm−1, G) = jxi1 ,··· ,xim (f1, · · · , fm−1, fm)
Hence we get
deg jxi1 ,··· ,xim (f1, · · · , fm−1, fm) + deg
≤ d1 + · · ·+ dm−1 −m+ degG
deg ∂G
+ dm − (d1 + · · ·+ dm−1 + dm −m) + deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG .
In particular, when the maximum is realized,
+dm−(d1+· · ·+dm−1+dm−m)+ max
1≤i1,··· ,im≤n
deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG
+dm−(d1+· · ·+dm−1+dm−m− max
1≤i1,··· ,im≤
deg jxi1 ,··· ,xim (f1, · · · , fm)) ≤ degG
+ dm −∇ ≤ degG.
In order to state our main theorem one needs to fix some more notations: we
denote p̄ the leading term of a polynomial p ∈ K[x1, · · · , xn] and for any subalgebra
A ⊂ K[x1, · · · , xn], we denote gr(A) := K[Ā] the subalgebra generated by Ā =
{ā|a ∈ A}. We define si, ∀1 ≤ i ≤ m, as the degree of the minimal, if any,
polynomial of f̄i over Frac( gr(K[fj ]j 6=i)), the field of fractions of the subalgebra
generated by K[fj]j 6=i = K[f1, · · · , fi−1, fi+1, · · · , fm] and as +∞ otherwise. We
denote ⌊α⌋ the integral part of a real number α and agree that k/∞ = 0 when
0 ≤ k < ∞.
Theorem. Let G be a polynomial in K[f1, · · · , fm]. Then the following minoration
holds, ∀1 ≤ i ≤ m,
degG ≥ di · degfi G−∇ · ⌊
degfi G
Proof. It is of course sufficient to prove it for i = m. First remark that a polynomial
m ∈ A[fm], where gi ∈ A := K[f1, · · · , fm−1], has degree strictly smaller
than maxi deg gi + i · dm if and only if
Ĝ :=
deg gi+i·dm=max
ḡif̄
m = 0
so if sm = +∞, which means such an annihilation cannot occur, then the minoration
in the Theorem is clear. Let’s assume now that f̄m does have a minimal polynomial
p(f̄m) = 0 with p = p(X) ∈ F [X ] where F is the field of fractions of gr(A) and X a
new indeterminate (whence sm := degX p). The following easy lemma constitutes
the very improvement with respect to [3]: it simplifies the proof a lot, makes it
A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES3
more general and even stronger in the sense that one does not need the estimate
(1) anymore.
Lemma. Let G =
m be in A[fm] and
h(X) :=
deg gi+i·dm=max
i ∈ gr(A)[X ] ( hence Ĝ = h(f̄m)).
If degG < degfm G ·dm then Ĝ = 0 or, equivalently, h(X) ∈ (p(X)) := p(X) ·F [X ].
Moreover if h′(X) 6= 0, where h′ is the derivative of h, then ∂̂G
= h′(f̄m) and more
generally, while h(k) 6= 0, one has ∂̂
= h(k)(f̄m).
Proof. If degG < degfm G · dm then degG < maxi deg gi + i · dm and Ĝ = 0 as
already remarked above.
Assume that h′ 6= 0. One has
ḡif̄
m = h(f̄m) where I := {i| deg gi + i · dm ≥ deg gj + j · dm ∀j} and
iḡif̄
m where I
′ := {i| deg igi + (i− 1) · dm ≥ deg jgj + (j − 1) · dm ∀j}.
It remains to notice that I ′ = I ∩ N∗ when this intersection is not empty, which
occurs exactly when h′ 6= 0. �
Let now k be the maximal number such that h(X) ∈ (p(X)k). Clearly degfm G ≥
deg h ≥ k · deg p = ksm hence k ≤ ⌊
degfm G
⌋. One has h(k) /∈ (p(X)) hence, by the
Lemma,
≥ dm · degfm
= dm · (degfm G− k)
and, by property (2),
degG ≥ dm · (degfm G− k) + k · dm − k · ∇ = dm · degfm G− k · ∇.
A straightforward computation gives the following
Corollary 1. Define, ∀i = 1, · · · ,m, Ni = Ni(f1, · · · , fm) := sidi − ∇. Let G be
a polynomial in K[f1, · · · , fm] and, ∀i = 1, · · · ,m, let degfi G = qisi + ri be the
euclidean division of degfi G by si. Then the following minoration holds
degG ≥ qi ·Ni + ridi.
The special case m = 2 corresponds to the main result of [3] (where s1, s2 are
easy to compute):
Corollary 2. If m = 2, σi :=
gcd(d1,d2)
with (i, j) = (1, 2) and (2, 1) and N :=
σ1d1 −∇ = σ2d2 −∇ then the following minoration holds, for i = 1, 2,
degG ≥ qi ·N + ridi
where degfi G = qisi + ri.
Proof. Let us prove it for i = 2. By corollary 1 it suffices to prove that s2 ≥ σ2
1: s2
is the degree of the minimal polynomial of f2 over Frac( gr(K[f1])) = Frac(K[f̄1]) =
K(f̄1):
p(f̄2) = f̄
2 + ps2−1(f̄1)f̄
2 + · · ·+ p1(f̄1)f̄2 + p0(f̄1) = 0(3)
1Actually equality holds, as proved in [3] using Zaks Lemma, it is however possible to show it
easily and without this result.
4 S. VÉNÉREAU
hence ∃0 ≤ i 6= j ≤ s2 such that deg pi(f̄1)f̄
2 = deg pj(f̄1)f̄
2 . It follows that
i · d2 ≡ jd2 mod d1 whence d1 | (i − j)d2 and i − j ∈ Z
gcd(d1,d2)
which gives
s2 ≥ |i− j| ≥
gcd(d1,d2)
= σ2. �
Corollary 3. Let G be a polynomial in K[f1, · · · , fm] such that degG = 1. Then,
∀i = 1, · · · ,m, degfi G = 0 or di = 1 or Ni = sidi −∇ ≤ 1.
Proof. Otherwise, by corollary 1, degG = 1 ≥ qiNi + ridi ≥ min{Ni, di} ≥ 2, a
contradiction. �
Corollary 4. Assume m = n and K[f1, · · · , fn] = K[x1, · · · , xn] i.e. f1, · · · , fn
define an automorphism (well-known fact). Then ∀i = 1, · · · , n, di = 1 or sidi ≤
d1 + · · · + dn − n + 1. In particular, if dmax ≥ dj , ∀j, and dmax ≥ 2 (i.e. the
automorphism is not affine) then smax ≤ n− 1.
Proof. One has ∇ = ∇(f1, · · · , fn) = d1+ · · ·+dn−n−deg jx1,··· ,xn(f1, · · · , fn) =
d1+ · · ·+dn−n. Moreover ∀j = 1, · · · , n, there exists Gj ∈ K[f1, · · · , fn] such that
xj = Gj and, ∀i = 1, · · · , n, degfi Gj ≤ 1 for at least one j = 1, · · · , n otherwise
K[f1, · · · , fj−1, fj+1, · · · , fm] = K[x1, · · · , xn] which is impossible. Whence, by
corollary 3, di = 1 or sidi ≤ ∇ + 1 = d1 + · · · + dn − n + 1. With dmax one gets
smaxdmax ≤ d1 + · · · + dn − n + 1 ≤ ndmax − n + 1 ≤ ndmax − 1 (n ≥ 2) and it
follows that smax ≤ n− 1. �
Corollary 5 (Tameness Theorem in dimension two). Every automorphism of K[x1, x2]
is tame i.e. a product of affine and elementary ones. Recall that an automorphism
τ : K[x1, x2] → K[x1, x2] is called elementary when, up to exchanging x1 and x2,
τ(x1) = x1 + p(x2) and τ(x2) = x2 for some p(X) ∈ K[X ].
Proof. Let α : K[x1, x2] → K[x1, x2] be an automorphism defined by α(xi) = fi
for i = 1, 2. We prove the corollary by induction on d1 + d2 = deg f1 + deg f2.
If d1 + d2 = 2 then d1 = d2 = 1 and α is affine.
Assume d1 + d2 ≥ 3. Without loss of generality d1 ≤ d2 and d2 ≥ 2 whence,
by corollary 4, s2 = 1 and the relation (3) in the proof of corollary 2 becomes:
f̄2 = p(f̄1) where p(X) must be of the form p(X) = ps1X
s1 ∈ K[X ]. Taking the
elementary automorphism τ defined τ(x1) = x1 and τ(x2) = x2 − p(X) one has a
new pair f ′1 := ατ(x1) = α(x1) = f1 and f
2 := ατ(x2) = α(x2 − p(X)) = f2− p(f1)
with degrees d′1 = d1 and d
2 < d2 hence d
2 < d1+d2. By induction ατ is tame
and so is α. �
References
[1] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant and applications,
preprint.
[2] J.T. Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl.
221, 1995, 19–29. MR 96c:14014
[3] Shestakov, Ivan P. and Umirbaev, Ualbai U., Poisson brackets and two-generated subalgebras
of rings of polynomials, J. Amer. Math. Soc. 17(1), 2004, 181–196 (electronic).
Stéphane Vénéreau
Mathematisches Institut
Universität Basel
Rheinsprung 21, CH-4051 Basel
Switzerland
stephane.venereau@unibas.ch
	References
