# Finitely generated bimodules over Weyl algebras

Niels Lauritzen and Jesper Funch Thomsen

###### Abstract

Let $A$ be the $n$-th Weyl algebra over a field of characteristic zero, and
$\varphi:A\rightarrow A$ an endomorphism with $S=\varphi(A)$. We prove that if
$A$ is finitely generated as a left or right $S$-module, then $S=A$. The proof
involves reduction to large positive characteristics. By holonomicity, $A$ is
always finitely generated as an $S$-bimodule. Moreover, if this bimodule
property could be transferred into a similar property in large positive
characteristics, then we could again conclude that $A=S$. The latter would
imply the Dixmier Conjecture.

## Introduction

Let $\varphi:A\rightarrow A$ denote an endomorphism of the $n$-th Weyl algebra
over a field $k$. Let $S=\varphi(A)$ denote the image of $\varphi$, and
consider $A$ as a bimodule over $S$ by left and right multiplication. When $k$
is a field of characteristic zero, it is known [5] that $A$ is holonomic as an
$S$-bimodule (see [14] for an interpretation of this result using tensor
products of bimodules), and therefore, $A$ is finitely generated (in fact,
cyclic) as an $S$-bimodule. In positive characteristic, the Weyl algebra is an
Azumaya algebra, and thus, finite generation of $A$ as an $S$-bimodule is
actually equivalent to finite generation of $A$ as an $S$-module from either
left or right. In large characteristics, this would imply that $\varphi$
induces a finite étale endomorphism of the center of $A$, which leads to the
conclusion that $A=S$.

Therefore, it becomes interesting to transfer the finite generation of $A$ as
an $S$-bimodule in characteristic zero to large positive characteristics. This
would imply $A=S$ in characteristic zero, providing a proof of Dixmier’s
conjecture also resolving the Jacobian conjecture [4, p. 297], [10], [17],
which is a long standing open problem [11, p. 301]. Regrettably, the direct
deduction of finite generation of $A$ as an $S$-bimodule in large positive
characteristics from its finite generation in characteristic zero remains
elusive. Nevertheless, the condition of being finitely generated from either
left or right translates nicely to large positive characteristics.
Consequently, if $A$ is finitely generated as an $S$-module from either left
or right, we can conclude $A=S$ in characteristic zero. This result answers
the last question posed in [13] affirmatively.

To make the positive characteristic reduction work, a suitable version of the
simply connectedness of affine space in positive characteristic is needed (see
Theorem 4.6)111Michel Brion has told us that a refinement of the suitable
positive characteristic version of simply connectedness follows from a
computation of the prime to $p$ fundamental group of affine space using
results by Orgogozo. We will address this in [15], where Weyl algebra
endomorphisms are treated exclusively in positive characteristics.. This
involves reduction to characteristic zero and occupies Section 4 in the last
part of the paper. Section 4 does not depend on the previous sections and can
be read separately. Along the way, we give a very simple proof (see Theorem
4.3) of the result [4, (1.4) Corollary] that if $\varphi$ is an automorphism
of affine $n$-space, then $\deg(\varphi^{-1})\leq\deg(\varphi)^{n-1}$.

In the end of the paper we sketch how degree bounds for Gröbner bases also
lead to the simply connectedness in positive characteristic.

The most accessible way to handle transitions between characteristic zero and
positive characteristic in our setup is through ultraproducts. We begin by
briefly introducing this approach.

##  1 Ultraproducts

In this section $X$ denotes a non-empty set and $R$ a commutative ring.

###  1.1 Filters

A _filter_ (see also [8, Chapter I, §6]) on $X$ is a subset
$\mathcal{F}\subset 2^{X}$ with $\emptyset\not\in\mathcal{F}$, $A\cap
B\in\mathcal{F}$ if $A,B\in\mathcal{F}$ and $B\in\mathcal{F}$ if
$A\in\mathcal{F}$ and $B\supset A$, where $A$ and $B$ are subsets of $X$.

A maximal filter is called an _ultrafilter_. The set
$\mathcal{F}_{a}=\\{S\subset X\mid a\in S\\}$ of all subsets of $X$ containing
a specific $a\in X$ is an ultrafilter on $X$. Such a filter is called
_principal_. The properties below follow (almost) immediately from the
definition of a filter.

  1. (i)

Every filter is contained in an ultrafilter (by Zorn’s lemma).

  2. (ii)

If $\mathcal{S}$ is a non-empty collection of subsets of $X$ and
$S_{1}\cap\cdots\cap S_{n}\neq\emptyset$ for finitely many
$S_{1},\dots,S_{n}\in\mathcal{S}$, then

$\\{Z\subset X\mid Z\supset S_{1}\cap\cdots\cap S_{n}\ \text{for finitely many
$S_{1},\dots,S_{n}\in\mathcal{S}$}\\}$

is a filter containing $\mathcal{S}$.

  3. (iii)

An ultrafilter $\mathcal{U}$ has the property that $A\in\mathcal{U}\text{or
}X\setminus A\in\mathcal{U}$ for every subset $A\subset X$.

All ultrafilters on finite sets are principal. On infinite sets there exists
non-principal ultrafilters. In fact,

  1. 4.

An ultrafilter contains the (filter of) cofinite subsets on an infinite set if
and only if it is non-principal.

###  1.2 Ultraproducts

Let $(S_{i})_{i\in X}$ be a family of sets. A filter
$\mathcal{F}\neq\emptyset$ on $X$ defines an equivalence relation on
$\prod_{i\in X}S_{i}$ given by $(s_{i})\sim(t_{i})$ if and only if $\\{i\mid
s_{i}=t_{i}\\}\in\mathcal{F}$. We use the notation

$\prod_{\mathcal{F}}S_{i}:=\left(\prod_{i\in X}S_{i}\right)\bigg{/}\sim.$

If $S_{i}$ are commutative rings, then
$I_{\mathcal{F}}=\\{(s_{i})\mid(s_{i})\sim 0\\}$ is an ideal in $\prod_{i\in
X}S_{i}$ and

$\prod_{\mathcal{F}}S_{i}=\prod_{i\in X}S_{i}\bigg{/}I_{\mathcal{F}}.$

Suppose that $S_{i}$ are fields. Then $\mathcal{F}\mapsto I_{\mathcal{F}}$
gives an inclusion preserving correspondence between filters on $X$ and proper
ideals in $\prod_{i\in X}S_{i}$. The filter corresponding to a proper ideal
$I$ is $\\{Z(s)\mid s\in I\\}$, where $Z(s)=\\{i\in X\mid s_{i}=0\\}$. In
particular, we get that $\prod_{\mathcal{U}}S_{i}$ is a field if $\mathcal{U}$
is an ultrafilter on $X$.

Let $\operatorname{Specm}(R)$ denote the set of maximal ideals of $R$ and
$\operatorname{Jac}(R)$ the Jacobson radical of $R$ i.e., the intersection of
all maximal ideals of $R$.

######  Proposition 1.1.

Let $R$ be an integral domain with $\operatorname{Jac}(R)=\\{0\\}$,
$X=\operatorname{Specm}(R)$ and $X_{r}=\\{\mathfrak{m}\in X\mid
r\notin\mathfrak{m}\\}$ for $r\in R$. Then

$\mathcal{F}=\\{Z\subseteq X\mid X_{r}\subseteq Z\text{ for some $r\in
R\setminus\\{0\\}$}\\}$

is a filter on $X$. Let $\mathcal{U}$ be an ultrafilter on $X$ containing
$\mathcal{F}$. Then $\prod_{\mathcal{U}}R/\mathfrak{m}$ contains the fraction
field of $R$.

###### Proof.

Notice that $X_{r}=\emptyset$ if and only if $r=0$ by the assumption
$\operatorname{Jac}(R)=\\{0\\}$. Therefore $\emptyset\notin\mathcal{F}$. Let
$Z_{1},Z_{2}\in\mathcal{F}$ with $X_{r}\subseteq Z_{1}$ and $X_{s}\subseteq
Z_{2}$, where $r,s\in R\setminus\\{0\\}$. Then $rs\in R\setminus\\{0\\}$ and
$X_{rs}\subseteq Z_{1}\cap Z_{2}$. Clearly, $Z_{2}\in\mathcal{F}$ if
$Z_{1}\in\mathcal{F}$ and $Z_{1}\subseteq Z_{2}$.

Let

$\pi:R\to\prod_{\mathcal{U}}R/\mathfrak{m}$

be the canonical map. Suppose that $r\in R\setminus\\{0\\}$ and $\pi(r)=0$.
Then $r\in\mathfrak{m}$ for every $\mathfrak{m}\in Z$ for some
$Z\in\mathcal{U}$ so that $Z\subseteq X\setminus X_{r}$. Therefore $X\setminus
X_{r}\in\mathcal{U}$ contradicting that $X_{r}\in\mathcal{U}$. It follows that
$\pi$ is injective and therefore that $\prod_{\mathcal{U}}R/\mathfrak{m}$
contains the fraction field of $R$. ∎

For $F\subset R[x_{1},\dots,x_{n}]$, we let $V_{S}(F)=\\{\alpha\in S^{n}\mid
f(\alpha)=0\text{ for all }f\in F\\}$, where $S$ is an $R$-algebra.

######  Lemma 1.2.

Let $\mathcal{F}$ be a filter on a set $X$. Consider
$F=\\{f_{1},\dots,f_{m}\\}\subset R[x_{1},\dots,x_{n}]$ and let

$S=\prod_{\mathcal{F}}S_{j},$

where $\\{S_{j}\mid j\in X\\}$ is a set of $R$-algebras. Fix elements
$\gamma_{i}=(\gamma_{i,j})_{j\in X}$ in $\prod_{j\in X}S_{j}$, for
$i=1,2,\dots,n$. Then $([\gamma_{1}],[\gamma_{2}],\dots,[\gamma_{n}])\in
V_{S}(F)$ if and only if there exists $B\in\mathcal{F}$ such that
$(\gamma_{1,j},\gamma_{2,j},\dots,\gamma_{n,j})\in V_{S_{j}}(F)$ for $j\in B$.

###### Proof.

Notice that if $f\in R[x_{1},\dots,x_{n}]$ is a polynomial, then we have the
following identity in $S$

$f([\gamma_{1}],[\gamma_{2}],\dots,[\gamma_{n}])=[\big{(}f(\gamma_{1,j},\gamma_{2,j},\dots,\gamma_{n,j})\big{)}_{j\in
X}].$ (1.2.1)

In particular, $f([\gamma_{1}],[\gamma_{2}],\dots,[\gamma_{n}])=0$ if and only
if there exists a $B\in\mathcal{F}$, such that
$f(\gamma_{1,j},\gamma_{2,j},\dots,\gamma_{n,j})=0$ for $j\in B$. Using that
$\mathcal{F}$ is stable under finite intersections, the claim is a direct
consequence of (1.2.1). ∎

##  2 The Weyl algebra

In the following $\mathbb{N}=\\{0,1,2\,\dots\\}$ denotes the natural numbers.
For $n\in\mathbb{N}$ and an $n$-tuple $a=(a_{1},\dots,a_{n})$ of elements in a
ring $A$, we use the notation $a^{v}$ with
$v=(v_{1},\dots,v_{n})\in\mathbb{Z}^{n}$ to denote

$a_{1}^{v_{1}}\cdots a_{n}^{v_{n}}$

if $v\in\mathbb{N}^{n}$ and $0$ if
$v\in\mathbb{Z}^{n}\setminus\mathbb{N}^{n}$. We will use $a\in\mathbb{Z}$ to
denote $(a,\dots,a)\in\mathbb{Z}^{n}$, when it fits the context. For integral
vectors $u=(u_{1},\dots,u_{n})$ and $v=(v_{1},\dots,v_{n})$, we let $u\leq v$
denote the partial order given by $u_{1}\leq v_{1},\dots,u_{n}\leq v_{n}$. For
$a,b\in A$, we let $[a,b]=ab-ba$.

Let $R$ denote a commutative ring. For $n\in\mathbb{N}$, $A_{n}(R)$ denotes
the $n$-th Weyl algebra over $R$. This is the (free) $R$-algebra generated by
$2n$ variables
$x=(x_{1},\dots,x_{n}),\partial=(\partial_{1},\dots,\partial_{n})$ with
relations

$\displaystyle\begin{split}[x_{i},x_{j}]&=0\\\
[\partial_{i},\partial_{j}]&=0\\\
[\partial_{i},x_{j}]&=\delta_{ij}\end{split}$ (2.0.1)

for $i,j=1,\dots,n$. Elements in

$S_{n}=\\{x^{u}\partial^{v}\mid u,v\in\mathbb{N}^{n}\\}\subset A_{n}(R)$

are called standard monomials. We have not been able to track down a proof of
the following result (for general commutative rings). We therefore outline a
proof (of linear independence) shown to us by J. C. Jantzen.

######  Proposition 2.1.

$A_{n}(R)$ is a free $R$-module with basis $S_{n}$.

###### Proof.

Using the relations (2.0.1) it follows that $S_{n}$ spans $A_{n}(R)$ as an
$R$-module. To prove linear independence of $S_{n}$, we let $A_{n}(R)$ act on
the free $R$-module with basis

$M=\\{X^{u}D^{v}\mid u,v\in\mathbb{N}^{n}\\}$

through

$\displaystyle x_{i}\cdot X^{u}D^{v}$ $\displaystyle=X^{u+e_{i}}D^{v}$
$\displaystyle\partial_{i}\cdot X^{u}D^{v}$
$\displaystyle=X^{u}D^{v+e_{i}}+u_{i}X^{u-e_{i}}D^{v}$

for $i=1,\dots,n$, where $e_{i}$ denotes the $i$-th canonical basis vector.
Suppose that

$\Delta=\sum_{u,v\in\mathbb{N}^{n}}c_{uv}x^{u}\partial^{v}=0$

in $A_{n}(R)$. Then acting on $M$ we get

$\Delta\cdot X^{0}D^{0}=\sum_{u,v\in\mathbb{N}^{n}}c_{uv}X^{u}D^{v}=0$

showing that $c_{uv}=0$ thereby proving the $R$-linear independence of
$S_{n}$. ∎

######  Definition 2.2.

The degree of $f\in A_{n}(R)\setminus\\{0\\}$, where

$f=\sum_{u,v\in\mathbb{N}^{n}}a_{uv}x^{u}\partial^{v}$

is defined as

$\deg(f)=\max\\{|u|+|v|\mid a_{uv}\neq 0\\},$

where

$|w|=w_{1}+\dots+w_{n}$

for $w=(w_{1},\dots,w_{n})\in\mathbb{N}^{n}$. The degree of an endomorphism
$\varphi:A_{n}(R)\rightarrow A_{n}(R)$ is defined as

$\deg(\varphi)=\max\\{\deg(\varphi(x_{1})),\dots,\deg(\varphi(x_{n})),\deg(\varphi(\partial_{1})),\dots,\deg(\varphi(\partial_{n}))\\}.$

###  2.1 The Weyl algebra in positive characteristic

######  Theorem 2.3.

Suppose that $\operatorname{char}(R)=p$, where $p$ is a prime number and let

$C=R[x_{1}^{p},\dots,x_{n}^{p},\partial_{1}^{p},\dots,\partial_{n}^{p}]\subset
A:=A_{n}(R).$

Then

  1. (i)

The center of $A$ is equal to $C$.

  2. (ii)

If $Q_{1},\dots,Q_{n},P_{1},\dots,P_{n}\in A$ satisfy the commutation
relations for the Weyl algebra i.e.,

$\displaystyle\begin{split}[Q_{i},Q_{j}]&=0\\\ [P_{i},P_{j}]&=0\\\
[P_{i},Q_{j}]&=\delta_{ij}\end{split}$

for $i,j=1,\dots,n$, then

$\\{Q^{\alpha}P^{\beta}\mid\alpha,\beta\in\mathbb{N}^{n},0\leq\alpha,\beta\leq
p-1\\}$

is a basis for $A$ as a module over $C$.

  3. (iii)

$A$ is an Azumaya algebra over $C$ and (therefore) $F(N)=N\otimes_{C}A$
defines an equivalence between the category of $C$-modules and the category of
$C$-linear $A$-bimodules (i.e., $A$-bimodules $M$ satisfying $rm=mr,$ for
$r\in C$ and $m\in M$) with inverse

$G(M)=M^{A}=\\{m\in M\mid am=ma,\forall a\in A\\}.$

Here $N$ denotes a $C$-module and $M$ a $C$-linear $A$-bimodule . The
equivalence preserves finitely generated modules.

###### Proof.

See [13, Theorem 1.7] for a proof of i and ii. Let
$A^{e}=A\otimes_{C}A^{{\operatorname{op}}}.$ To prove iii, we need to show
(cf. [12, Theorem III.5.1, 2)]) that the natural map

$\varphi:A^{e}\rightarrow\operatorname{End}_{C}(A)$ (2.1.1)

of $C$-algebras given by $\varphi(a\otimes b)(x)=axb$ is an isomorphism.
Notice that $A^{e}$ and $\operatorname{End}_{C}(A)$ are free $C$-modules of
rank $p^{4n}$. The elements $\partial_{1}\otimes 1,\dots,\partial_{n}\otimes
1,1\otimes\partial_{1},\dots,1\otimes\partial_{n}$ and $x_{1}\otimes
1,\dots,x_{n}\otimes 1,1\otimes x_{1},\dots,1\otimes x_{n}$ define
$a_{1},\dots,a_{2n},b_{1},\dots,b_{2n}\in A^{e}$ with
$[a_{i},a_{j}]=[b_{i},b_{j}]=0$ and $[a_{i},b_{j}]=\delta_{ij}$ for
$i,j=1,\dots,2n$.

As in the proof of Theorem 1.7$(ii)$ in [13], it follows that

$M=\\{\alpha^{u}\beta^{v}\mid u,v\in\mathbb{N}^{2n},0\leq u,v\leq
p-1\\}\subset\operatorname{End}_{C}(A)$

is a $C$-basis of $\operatorname{End}_{C}(A)$, where
$\alpha=(\varphi(a_{1}),\dots,\varphi(a_{2n}))$ and
$\beta=(\varphi(b_{1}),\dots,\varphi(b_{2n}))$. Therefore $\varphi$ is an
isomorphism of $C$-modules and $A$ is an Azumaya algebra over $C$. This
implies that $F$ and $G$ is an adjoint pair of inverse equivalences between
the category of $C$-modules and the category of $C$-linear $A$-bimodules by
[12, Theorem III.5.1, 3)]. Finally, finite generation is a categorical
property preserved under equivalences by [1, Proposition 21.8]. ∎

###  2.2 Endomorphisms of Weyl algebras

For an arbitrary ring $S$ and a ring endomorphism $\varphi:S\rightarrow S$, we
let ${}_{\varphi}S_{\varphi}$ denote the $S$-bimodule $S$ with multiplication

$xsy=\varphi(x)s\varphi(y),$

where $s,x,y\in S$. Similarly ${}_{\varphi}S$ denotes $S$ as a left module
with multiplication given by $xs=\varphi(x)s$. Using the notation from Theorem
2.3, we have the following corollary.

######  Corollary 2.4.

Let $\varphi:A\rightarrow A$ be an $R$-algebra homomorphism. Then

  1. (i)

$\varphi(C)\subset C$

  2. (ii)

$\varphi$ is an isomorphism if and only if $\varphi|_{C}$ is an isomorphism.

  3. (iii)

The bimodule ${}_{\varphi}A_{\varphi}$ is $C$-linear with
$({}_{\varphi}A_{\varphi})^{A}={}_{\varphi}C$. Furthermore,
${}_{\varphi}A_{\varphi}$ is finitely generated if and only if $\varphi|_{C}$
is finite.

###### Proof.

The first two claims are consequences of Theorem 2.3 ii with
$P_{i}=\varphi(\partial_{i})$ and $Q_{i}=\varphi(x_{i})$ for $i=1,\dots,n$.
For the proof of iii, we note that the $C$-linearity of
${}_{\varphi}A_{\varphi}$ follows from i and that
$({}_{\varphi}A_{\varphi})^{A}={}_{\varphi}C$ follows from Theorem 2.3ii. The
last statement in iii follows from Theorem 2.3iii. ∎

For completeness we include the proof of the following well known result (see
[6, Corollary 3.4] for the characteristic zero case).

######  Theorem 2.5.

Let $R$ be a reduced commutative ring. If $\varphi:A_{n}(R)\rightarrow
A_{n}(R)$ is an automorphism of $R$-algebras, then

$\deg(\varphi^{-1})\leq\deg(\varphi)^{2n-1}.$

###### Proof.

Using Proposition 2.1 and expanding
$\varphi(x_{i}),\varphi(\partial_{j}),\varphi^{-1}(x_{i}),\varphi^{-1}(\partial_{j})$
in the basis of the standard monomials for $i,j=1,\dots,n$, we may assume that
$R$ is finitely generated over $\mathbb{Z}$ and therefore a Jacobson ring (see
[7, Chapter V, §3.4]). Since $R$ is reduced, $\operatorname{Jac}(R)=0$ and
there exists a maximal ideal $\mathfrak{m}$ not containing the coefficient of
a highest degree standard monomial in the definition of $\deg(\varphi^{-1})$
(cf. Definition 2.2). But $k=R/\mathfrak{m}$ is a field of characteristic
$p>0$ and $\varphi,\varphi^{-1}$ induce inverse automorphisms
$\bar{\varphi},\bar{\varphi}^{-1}$ of $A_{n}(k)$ with
$\deg(\bar{\varphi}^{-1})=\deg(\varphi^{-1})$ and
$\deg(\bar{\varphi})\leq\deg(\varphi)$.

Now the result follows by restricting $\bar{\varphi}$ to the center $C$ of
$A_{n}(k)$ (cf. Corollary 2.4ii) and using Theorem 4.3 as
$\deg(\bar{\varphi}|_{C})=\deg(\bar{\varphi})$. ∎

######  Remark 2.6.

Notice that the bound in Theorem 2.5 only depends on $\deg(\varphi)$ and $n$.
In particular, it does not depend on the (reduced) ring $R$. We can only prove
Theorem 2.5 when $R$ is reduced. The independence of $R$ in bounding the
degree of the inverse in Theorem 2.5 for nilpotent rings should be equivalent
to the Dixmier conjecture in analogy with [3, Theorem (1.1)] and [4, Ch. I,
Prop. (1.2)].

The following result is non-trivial and crucial.

######  Lemma 2.7.

Let $k$ be a field of characteristic $p>0$ and $\varphi$ an endomorphism of
$A_{n}(k)$. Then $\varphi$ restricts to an étale endomorphism of the center of
$A_{n}(k)$ if $p>2\deg(\varphi).$

###### Proof.

This is a consequence of [17, Corollary 3.3]. ∎

######  Remark 2.8.

In [15], the bound in Lemma 2.7 is strengthened to $p>\deg(\varphi)$.

##  3 The main result

The following lemma is the positive characteristic version of Theorem 3.3,
which is our main result. It basically uses the equivalence between bimodules
and modules over the center (cf. Theorem 2.3iii) to reduce to the commutative
case of finite étale endomorphisms of polynomial rings, which is the focus of
section 4 and in particular Theorem 4.6. Notice again that it is crucial that
an endomorphism of the Weyl algebra restricts to an étale endomorphism of the
center if the characteristic is large enough (cf. Lemma 2.7).

######  Lemma 3.1.

Let $k$ be a field of characteristic $p>0$, $A=A_{n}(k)$ and
$\varphi:A\rightarrow A$ an endomorphism of degree $\leq d$. Then there exists
a uniform bound $D=D(n,d)$ only depending on $n$ and $d$, such that if $p>D$
and ${}_{\varphi}A_{\varphi}$ is finitely generated, then $\varphi$ is an
isomorphism.

###### Proof.

Let $C$ be the center of $A_{n}(k)$. By Lemma 2.7, $\varphi|_{C}$ is étale for
$p>2d$. The assumption that ${}_{\varphi}A_{\varphi}$ is finitely generated
shows that $\varphi|_{C}$ is finite by Corollary 2.4iii. Let
$D(n,d)=\max\\{2d,N(2n,d)\\}$, where $N(2n,d)$ refers to the uniform bound in
Theorem 4.6. Then $\varphi|_{C}$ is an isomorphism for $p>D(n,d)$ by Theorem
4.6 and finally we conclude that $\varphi$ is an isomorphism by Corollary
2.4ii. ∎

To go to characteristic zero we need the following characterization of (left)
generating sets.

######  Lemma 3.2.

Let $R$ be a commutative ring, $A=A_{n}(R)$, $\varphi:A\rightarrow A$ an
$R$-algebra endomorphism and $S=\varphi(A)$. Suppose that
$G=\\{g_{1},\dots,g_{m}\\}\subset A$ satisfies

  1. (i)

$1\in G$

  2. (ii)

There exists $s_{ijl},t_{ijl}\in S$ , such that

$\displaystyle g_{j}x_{i}$ $\displaystyle=s_{ij1}g_{1}+\cdots+s_{ijm}g_{m}$
(3.0.1) $\displaystyle g_{j}\partial_{i}$
$\displaystyle=t_{ij1}g_{1}+\cdots+t_{ijm}g_{m}$ (3.0.2)

for $l=1,\dots,m$ and $1\leq i,j\leq n$.

Then $G$ generates $A$ as a left $S$-module i.e.,

$A=Sg_{1}\cdots+Sg_{m}.$

Conversely if $G=\\{g_{1},\dots,g_{m}\\}$ is a generating set for $A$ as a
left $S$-module with $1\in G$, then $G$ satisfies (3.0.1) and (3.0.2) above.

###### Proof.

Let $M=Sg_{1}+\cdots+Sg_{m}$. We wish to prove that $M=A$. Using (3.0.1) we
get that $x^{u}\in M$ for every $u\in\mathbb{N}^{n}$. Building on this, one
shows using (3.0.2) that $x^{u}\partial^{v}\in M$ for every
$u,v\in\mathbb{N}^{n}$. Therefore we must have $M=A$. The converse follows by
definition of a (left) generating set. ∎

######  Theorem 3.3.

Let $K$ be a field of characteristic $0$, $A=A_{n}(K)$ and
$\varphi:A\rightarrow A$ a $K$-algebra endomorphism with $S=\varphi(A)$. If
$A$ is finitely generated as a left or right module over $S$, then $S=A$.

###### Proof.

Assume that $A$ is generated by $G=\\{g_{1},\dots,g_{m}\\}$ as a left
$S$-module (the case of a right $S$-module is similar) with $1\in G$. We start
by fixing a finitely generated $\mathbb{Z}$-algebra $R^{\prime}$ of $K$
generated by the coefficients of $\varphi(x_{i}),\varphi(\partial_{i})$ for
$i=1,\dots,n$ in the basis of standard monomials (cf. Definition 2.2). In the
same way we extend $R^{\prime}$ to a finitely generated $\mathbb{Z}$-algebra
$R$, such that the coefficients of $g_{1},\dots,g_{m}$ and $s_{ijk},t_{ijk}$
of Lemma 3.2 are in $R$. For $\mathfrak{m}\in\operatorname{Specm}(R)$,
$R/\mathfrak{m}$ is a field of characteristic $p>0$ and it follows by
reduction modulo $\mathfrak{m}$ of (3.0.1) and (3.0.2) that
${}_{\bar{\varphi}}A_{n}(R/\mathfrak{m})_{\bar{\varphi}}$ is finitely
generated as a left module and therefore as a bimodule, where
$\bar{\varphi}:A_{n}(R/\mathfrak{m})\rightarrow A_{n}(R/\mathfrak{m})$ denotes
the reduction of $\varphi$ modulo $\mathfrak{m}$.

Now let $D=D(n,\deg(\varphi))$ be the bound from Lemma 3.1. Let $M$ be the
product of the prime numbers $\leq D$. Fix an ultrafilter $\mathcal{U}$ on
$\operatorname{Specm}(R)$ as in Proposition 1.1. Then $\bar{\varphi}$ is an
isomorphism for every $\mathfrak{m}\in X_{M}$ (with the notation in
Proposition 1.1) by Lemma 3.1 and Theorem 2.5 along with Lemma 1.2 show that
$\varphi$ is an isomorphism over the ultraproduct
$\prod_{\mathcal{U}}R/\mathfrak{m}$. Therefore $\varphi$ is an isomorphism
over $K$ by [13, Lemma 3.3]. ∎

The last part of the paper concerns the proof of Theorem 4.6 below, which is
the key result needed in the proof of Lemma 3.1 (and therefore Theorem 3.3)
above.

##  4 Bounds on polynomial endomorphisms

In this section we deal exclusively with commutative rings and introduce new
notation independent of the notation in the previous sections i.e., $A$ no
longer refers to the Weyl algebra.

Let $B=K[x_{1},\dots,x_{n}]$ be the polynomial ring of $n$ variables over an
arbitrary field $K$ (with $n>0$), $\varphi:B\rightarrow B$ a $K$-algebra
homomorphism given by $\varphi(x_{i})=f_{i}$ and $\varphi(B)=:A\subset B$.
Denote the fields of fractions of $A$ and $B$ by $K(A)$ and $K(B)$
respectively. We let

$\deg(\varphi)=\max\\{\deg(f_{1}),\dots,\deg(f_{n})\\}.$ (4.0.1)

where $\deg(f_{i})$ denotes the total degree of $f_{i}$. For $d\in\mathbb{N}$
we let $B_{\leq d}$ denote the $K$-subspace of $B$ consisting of polynomials
$f$ with $\deg(f)\leq d$.

######  Lemma 4.1.

Let $d,r\in\mathbb{N}$, with $r>0$, and $h_{1},h_{2},\dots,h_{r}\in B_{\leq
d}$. Suppose that $m>n+d$ is an integer and that $h_{1},\dots,h_{r}$ do not
satisfy a non-trivial linear relation

$\varphi(b_{1})h_{1}+\cdots+\varphi(b_{r})h_{r}=0$

with $b_{i}\in B_{\leq m}$ for $i=1,\dots,r$. Then

$m\,(r-\deg(\varphi)^{n})<2^{n}\deg(\varphi)^{n-1}(n+d).$

###### Proof.

Notice first that $\deg(\varphi)>0$ with the given assumptions. Let $V$ denote
the subspace

$V=\left\\{\varphi(b_{1})h_{1}+\cdots+\varphi(b_{r})h_{r}\,\,\bigg{|}\,\,b_{1},\dots,b_{r}\in
B_{\leq m}\right\\}\subseteq B_{\leq m\deg(\varphi)+d}.$

By assumption

$\dim_{K}(V)=r\,\dim_{K}(B_{\leq m})\leq\dim_{K}(B_{\leq
m\cdot\deg(\varphi)+d}).$

Therefore

$r\frac{m^{n}}{n!}\leq
r\binom{m+n}{n}\leq\binom{m\,\deg(\varphi)+d+n}{n}\leq\frac{(m\,\deg(\varphi)+d+n)^{n}}{n!}$

showing that

$rm^{n}\leq(m\,\deg(\varphi)+d+n)^{n}<(m\,\deg(\varphi))^{n}+2^{n}(m\,\deg(\varphi))^{n-1}(d+n),$
(4.0.2)

where the last inequality follows by the assumption $d+n<m$ and as
$\deg(\varphi)>0$. The claimed inequality now follows from (4.0.2). ∎

In the following we will assume that $f_{1},f_{2},\dots,f_{n}$ are
algebraically independent over $K$. The induced field extension $K(A)\subseteq
K(B)$ is then finite.

######  Proposition 4.2.

$[K(B):K(A)]\leq\deg(\varphi)^{n}$

###### Proof.

Let $h_{1},h_{2},\dots,h_{r}$ denote a basis for $K(B)$ as a vector space over
$K(A)$. Clearing denominators we may assume that each $h_{i}$ is contained in
$B$. Choose $d$ such that $h_{i}\in B_{\leq d}$, for each $i$. Applying Lemma
4.1 we then conclude that

$(r-\deg(\varphi)^{n})\cdot m<2^{n}\deg(\varphi)^{n-1}(n+d),$

for every positive integer $m>n+d$. Therefore $r-\deg(\varphi)^{n}\leq 0$ and
$r\leq\deg(\varphi)^{n}$. ∎

###  4.1 The classical bound on the degree of the inverse

We will give a short and elementary proof of the following classical result
inspired by ideas in [18].

######  Theorem 4.3 ([4], Thm. 1.5).

If $\varphi$ is an automorphism of $B$, then
$\deg(\varphi^{-1})\leq\deg(\varphi)^{n-1}$.

###### Proof.

Suppose that $\varphi^{-1}(x_{i})=g_{i}$ for $i=1,\dots,n$. We may assume that
$K$ is an infinite field. Thus by a linear change of coordinates we can
assume, that each $g_{i}$ is a monic polynomial in $x_{n}$ with coefficients
in the subring $B^{\prime}=K[x_{1},x_{2},\dots,x_{n-1}]$ of $B$. Let $N$
denote the degree of $\varphi^{-1}$ and assume that $\deg(g_{n})=N$. Now
identify $B^{\prime}$ with the quotient ring $B/(x_{n})$ and use the notation
$\overline{h}$ for the image of $h\in B$ in $B/(x_{n})$.

Since $\varphi$ is an automorphism, $B=K[f_{1},f_{2},\dots,f_{n}]$ and
$B^{\prime}=K[\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n}}]$. We
claim that $\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}}$ are
algebraically independent over $K$ and that

$[K(x_{1},x_{2},\dots,x_{n-1}):K(\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}})]=\deg(\varphi^{-1}).$

By Proposition 4.2 this will end the proof, as
$\deg(\overline{f_{i}})\leq\deg(\varphi)$. Write

$g_{n}=a_{0}+a_{1}x_{n}+a_{2}x_{n}^{2}+\cdots+a_{N-1}x_{n}^{N-1}+x_{n}^{N}$

with $a_{i}\in B^{\prime}$. Then

$x_{n}=\varphi(g_{n})=\varphi(a_{0})+\varphi(a_{1})f_{n}+\cdots+\varphi(a_{N-1})f_{n}^{N-1}+f_{n}^{N},$

and

$c_{0}+c_{1}\cdot\overline{f_{n}}+\cdots+c_{N-1}\cdot\overline{f_{n}}^{N-1}+\overline{f_{n}}^{N}=0.$
(4.1.1)

where $c_{i}=\overline{\varphi(a_{i})}$ are elements in
$A^{\prime}=K[\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}}]$. It
follows that $\overline{f_{n}}$ is integral over $A^{\prime}$ and therefore
that $B^{\prime}=K[\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n}}]$
is integral over $A^{\prime}$. We conclude, that the elements
$\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}}$ are algebraically
independent and that
$[K(x_{1},x_{2},\dots,x_{n-1}):K(\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}})]$
equals the degree of the minimal polynomial of $\overline{f_{n}}$ over
$K(\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}})$. Consider the
polynomial

$F=c_{0}+c_{1}\cdot X+\cdots+c_{N-1}\cdot X^{N-1}+X^{N}\in A^{\prime}[X].$

By equation (4.1.1) we see that $\overline{f_{n}}$ is a root of $F$. So it
suffices to prove that $F$ is irreducible in $A^{\prime}[X]$ (and thus also in
$K(\overline{f_{1}},\overline{f_{2}},\dots,\overline{f_{n-1}})[X]$). To see
this observe first that $F$ is the image of $g_{n}$ under the $K$-algebra
automorphism $B\rightarrow A^{\prime}[X]$ which maps $x_{n}$ to $X$ and
$x_{i}$ to $\overline{f_{i}}$, for $i<n$. Then use that $g_{n}$ is irreducible
as $\varphi$ is an automorphism. ∎

###  4.2 Integral extensions

In the statement below we still work under the assumption that
$f_{1},f_{2},\dots,f_{n}$ are algebraically independent.

######  Proposition 4.4.

Let $g$ denote an element in $B$ of degree $D$ which is integral over $A$, and
let

$G=a_{0}+a_{1}T+a_{2}T^{2}+\cdots+a_{r-1}T^{r-1}+T^{r}\in K(A)[T],$

denote its minimal polynomial. Then $a_{i}\in\varphi(B_{\leq m})$ for
$i=0,\dots,r-1$, where

$m=2^{n}\deg(\varphi)^{n-1}(n+D\cdot\deg(\varphi)^{n}).$

###### Proof.

Applying Lemma 4.1 with $r=\deg(\varphi)^{n}+1,d=D\deg(\varphi)^{n}$ and $m$
as above, we conclude that there exists $b_{i}\in B_{\leq m}$, for
$i=0,1,\dots,\deg(\varphi)^{n}$ with

$\varphi(b_{0})+\varphi(b_{1})g+\cdots+\varphi(b_{\deg(\varphi)^{n}})g^{\deg(\varphi)^{n}}=0,$

such that not all $b_{i}$ are zero. The polynomial

$F=\varphi(b_{0})+\varphi(b_{1})T+\cdots+\varphi(b_{\deg(\varphi)^{n}})T^{\deg(\varphi)^{n}}\in
A[T]$

is thus divisible by $G$ i.e., $F=G\cdot H$ for some polynomial $H$ in
$K(A)[T]$. By [2, Proposition 5.15], $G\in A[T]$. Therefore we may even
conclude that $H\in A[T]$. Consider now $F,G$ and $H$ as polynomials in
$f_{1},f_{2},\dots f_{n}$ with coefficients in $K[T]$. Then the (total) degree
of $F$ is $\leq m$ which implies, that the same is true for its divisor $G$.
This ends the proof. ∎

Recall that the ring homomorphism $\varphi:B\rightarrow B$ is called integral
if $B$ is integral over the image $A=\varphi(B)$. In the present setup
$\varphi$ can only by integral if $f_{1},f_{2},\dots,f_{n}$ are algebraically
independent. Combining this observation with Proposition 4.2 and Proposition
4.4 above we find:

######  Theorem 4.5.

Let $m=2^{n}\deg(\varphi)^{n-1}(n+\deg(\varphi)^{n})$. The following
statements are equivalent.

  1. (i)

The ring homomorphism $\varphi:B\rightarrow B$ is integral.

  2. (ii)

There exists nonzero monic polynomials $F_{i}\in A[T]$, for $i=1,2,\dots,n$,
of degree $\deg(\varphi)^{n}$ and with coefficients in $\varphi(B_{\leq m})$,
such that $F_{i}(x_{i})=0$.

###### Proof.

The second statement clearly implies the first statement. So assume that
$A\subseteq B$ is an integral extension. By Proposition 4.4 we know that the
minimal polynomial $G_{i}$ for $x_{i}$ has coefficients in $\varphi(B_{\leq
m})$ for each $i$. Moreover, by Proposition 4.2 these minimal polynomials have
degree $\leq\deg(\varphi)^{n}$. Multiplying $G_{i}$ by a suitable power of $T$
we obtain a polynomial $F_{i}$ of degree $\deg(\varphi)^{n}$ which satisfies
the statement in (ii). ∎

The result below is a key component in the proof of Lemma 3.1 used in the
proof of Theorem 3.3.

######  Theorem 4.6.

Let $n,D\in\mathbb{N}$ be positive. There exists a uniform bound
$N(n,D)\in\mathbb{N}$ only depending on $n$ and $D$, such that when $k$ is a
field of characteristic $p>N(n,D)$ and $\varphi$ is a finite and étale
$k$-algebra endomorphism of the polynomial ring $k[x_{1},\dots,x_{n}]$ of
degree $\leq D$, then $\varphi$ is an automorphism.

###### Proof.

Suppose that the theorem is wrong. Then we may find an infinite sequence of
primes $p_{1}<p_{2}<\dots$ and corresponding fields $k_{1},k_{2},\dots$ of
characteristic $\operatorname{char}(k_{i})=p_{i}$ together with finite and
étale $k_{i}$-algebra endomorphisms $\varphi_{i}$ of
$k_{i}[x_{1},\dots,x_{n}]$ of degree $\leq D$, which are not automorphisms.
For simplicity we may assume that the determinant of the Jacobian matrix of
each $\varphi_{i}$ equals $1$.

Choose a non-principal ultrafilter $\mathcal{U}$ on the set
$\mathcal{P}=\\{p_{1},p_{2},\dots\\}$ and let $C$ denote the ultraproduct
$\prod_{\mathcal{U}}k_{i}$. Let further

$\varphi:C[x_{1},\dots,x_{n}]\rightarrow C[x_{1},\dots,x_{n}]$

denote the $C$-algebra endomorphism induced by $\varphi_{i},i>0$. We claim
that $\varphi$ is a finite and étale. That $\varphi$ is étale follows as the
determinant of the Jacobian matrix of each $\varphi_{i}$, and hence also of
$\varphi$, equals $1$. To conclude that $\varphi$ is finite we apply Theorem
4.5. It suffices to prove that each $x_{j}$ is integral over the image of
$\varphi$. By assumption $x_{j}$ is integral over the image of $\varphi_{i}$,
so by Theorem 4.5 we may find polynomials $p_{i,r}\in
k_{i}[x_{1},\dots,x_{n}]$, for $r=1,2,\dots,D^{n}$, of degree $\leq
m:=2^{n}D^{n-1}(n+D^{n})$, such that $x_{j}$ is a root of the monic polynomial

$P_{i}(T)=T^{D^{n}}+\sum_{r=1}^{D^{n}}\varphi_{i}(p_{i,r})T^{r-1}.$

Let now $p_{r}\in C[x_{1},\dots,x_{n}]$ denote the polynomial defined by the
collection of polynomials $(p_{i,r})_{i>0}$. Then $x_{j}$ is a root of the
monic polynomial

$P(T)=T^{D^{n}}+\sum_{r=1}^{D^{n}}\varphi(p_{r})T^{r-1},$

and we conclude that $\varphi$ is finite.

Next observe that $C$ is a field of characteristic zero and thus $\varphi$ is
an automomorphism by [4, (2.1) Theorem, (d)] with inverse $\varphi^{-1}$ of
some degree $D^{\prime}$. Fix, for each $i>0$, a lift $\psi_{i}$ of
$\varphi^{-1}$ to a $k_{i}$-algebra endomorphism of $k_{i}[x_{1},\dots,x_{n}]$
of degree $\leq D^{\prime}$. In this way $\varphi^{-1}$ is defined from
$\psi_{i}$, for $i>0$, in the same way as $\varphi$ was defined from
$\varphi_{i}$, for $i>0$. That $\psi_{i}$ is an inverse to $\varphi_{i}$ is
then a polynomial condition in the coefficients of $\psi_{i}$ and
$\varphi_{i}$. These polynomial conditions are satisfied for $\varphi^{-1}$
and $\varphi$, and thus $\psi_{i}$ and $\varphi_{i}$ will be mutually inverses
for every prime $p_{i}$ in some element $B$ in $\mathcal{U}$ by Lemma 1.2.
This is a contradiction as none of the endomorphisms $\varphi_{i}$, $i>0$, are
automorphisms. ∎

###  4.3 A Gröbner basis approach

Here we outline an alternative approach to obtaining bounds similar to the
ones in Theorem 4.5. We keep the notation of the previous section.

It suffices to prove that if $f_{1},\dots,f_{n}\in K[x_{1},\dots,x_{n}]$ are
polynomials of degree $\leq d$, then $x_{i}$ is integral over
$K[f_{1},\dots,f_{n}]$ for $i=1,\dots,n$ if and only if the minimal polynomial
for each $x_{i}$ has the form

$T^{m}+a_{m-1}(f_{1},\dots,f_{n})T^{m-1}+\cdots+a_{1}(f_{1},\dots,f_{n})T+a_{0}(f_{1},\dots,f_{n})=0,$
(4.3.1)

for $m\leq M(n,d)$ and $\deg(a_{j})\leq D(n,d)$ for $j=0,\dots,m-1$, where
$M(n,d),D(n,d)\in\mathbb{N}$ depend only on $n$ and $d$ (and not on $K$).

We will need the following degree bound on Gröbner bases for arbitrary term
orderings.

######  Theorem 4.7 ([9]).

Let $K$ be a field and $I\subset K[x_{1},\dots,x_{n}]$ an ideal generated by
polynomials of degree $\leq d$. Then there exists a Gröbner basis $G$ of $I$
with respect to any term order, such that

$\deg(f)\leq 2\left(\frac{d^{2}}{2}+d\right)^{2^{n-1}}$

for $f\in G$.

Combined with [2, Proposition 5.15], the existence of the degree bounded
minimal polynomials in (4.3.1) now follows from the lemma below.

######  Lemma 4.8 ([16]).

Suppose that $f,f_{1},\dots,f_{m}\in K[x_{1},\dots,x_{n}]$ and let

$I=(t-f,t_{1}-f_{1},\dots,t_{m}-f_{m})\subset
K[t,t_{1},\dots,t_{m},x_{1},\dots,x_{n}].$

Then

  1. (a)

$F(f,f_{1},\dots,f_{m})=0\qquad\iff\qquad F\in I\cap K[t,t_{1},\dots,t_{m}]$

for $F\in K[t,t_{1},\dots,t_{m}]$.

  2. (b)

Let $G$ be a Gröbner basis of $I$ with the lexicographic order
$x_{1}>\cdots>x_{n}>t>t_{1}>\cdots>t_{m}$. If $f$ is algebraic over
$K(f_{1},\dots,f_{m})$, then the minimal polynomial of $f$ over
$K(f_{1},\dots,f_{m})$ can be identified with the polynomial in

$G\cap\left(K[t,t_{1},\dots,t_{m}]\setminus K[t_{1},\dots,t_{m}]\right)$

with smallest leading term.

###### Proof.

For $F\in K[t,t_{1},\dots,t_{m}]$,

$F(f,f_{1},\dots,f_{m})=0\qquad\iff\qquad F\in I\cap K[t,t_{1},\dots,t_{m}]$

is a consequence of the division algorithm. The (elimination) ideal $I\cap
K[t,t_{1},\dots,t_{m}]$ is generated by $G\cap K[t,t_{1},\dots,t_{m}]$. The
result can now be deduced from the fact that the minimal polynomial (after
clearing denominators) can be identified with the polynomial containing $t$ in
$G$ with smallest leading term. ∎

## References

  * [1] F. W. Anderson and K. R. Fuller, _Rings and categories of modules_ , second ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992.
  * [2] M. Atiyah and I. Macdonald, _Introduction to commutative algebra_ , Addison-Wesley, Reading, 1969.
  * [3] H. Bass, _The Jacobian conjecture and inverse degrees_ , Arithmetic and geometry, Vol. II, Progr. Math, vol. 36, Birkhäuser, Boston, Mass., 1983, pp. 65–75.
  * [4] H. Bass, E. H. Connell, and D. Wright, _The Jacobian conjecture: reduction of degree and formal expansion of the inverse_ , Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330.
  * [5] V. V. Bavula, _A question of Rentschler and the Dixmier problem_ , Ann. of Math. (2) 154 (2001), no. 3, 683–702.
  * [6] , _The inversion formula for automorphisms of the Weyl algebras and polynomial algebras_ , J. Pure Appl. Algebra 210 (2007), no. 1, 147–159.
  * [7] N. Bourbaki, _Commutative algebra. Chapters 1–7_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989, Translated from the French, Reprint of the 1989 English translation.
  * [8] , _General topology. Chapters 1–4_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998, Translated from the French, Reprint of the 1989 English translation.
  * [9] T. Dubé, _The structure of polynomial ideals and Gröbner bases_ , SIAM J. Comput. 19 (1990), no. 4, 750–773.
  * [10] A. Kanel-Belov and M. Kontsevich, _The Jacobian conjecture is stably equivalent to the Dixmier conjecture_ , Mosc. Math. J. 7 (2007), no. 2, 209–218, 349.
  * [11] Ott-Heinrich Keller, _Ganze Cremona-Transformationen_ , Monatsh. Math. Phys. 47 (1939), no. 1, 299–306.
  * [12] M.-A. Knus and M. Ojanguren, _Théorie de la descente et algèbres d’Azumaya_ , Lecture Notes in Mathematics, vol. 389, Springer-Verlag, Berlin-New York, 1974.
  * [13] N. Lauritzen and J. F. Thomsen, _Two properties of endomorphisms of Weyl algebras_ , J. Algebra 479 (2017), 137–158.
  * [14] , _The graph of a Weyl algebra endomorphism_ , Bull. London Math. Soc. 53 (2021), 161–176.
  * [15] , _Étale covers and Weyl algebras in positive characteristics_ , (in preparation).
  * [16] D. Shannon and M. Sweedler, _Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence_ , J. Symbolic Comput. 6 (1988), 267–273.
  * [17] Y. Tsuchimoto, _Endomorphisms of Weyl algebra and $p$-curvatures_, Osaka J. Math. 42 (2005), no. 2, 435–452.
  * [18] Jie-Tai Yu, _Degree bounds of minimal polynomials and polynomial automorphisms_ , Journal of Pure and Applied Algebra 92 (1994), 199–201.