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\title{Holonomic modules and 1-generation in the Jacobian Conjecture}
\alttitle{Modules holonomes et 1-génération dans la conjecture jacobienne}

\author{\firstname{Volodymyr} \middlename{V.} \lastname{Bavula}}
\address{School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK}
\email{v.bavula@sheffield.ac.uk}

\begin{abstract}
Let $P_n$ be a polynomial algebra in $n$ indeterminates over a field $K$ of characteristic zero. An endomorphism $\s\in \End_K(P_n)$ is called a \emph{Jacobian map} if its Jacobian is a nonzero scalar. Each Jacobian map $\s$ is extended to an endomorphism $\s$ of the Weyl algebra $A_n$.

The \emph{Jacobian Conjecture} (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $\s$) $P_n$-\emph{module} $\sP_n$ is cyclic for all Jacobian maps $\s$. It is shown that the $A_n$-\emph{module} $\sP_n$ is cyclic for all Jacobian maps $\s$. Furthermore, the $A_n$-module $\sP_n$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_n$-module $\sP_n$ in terms of the degree $\deg (\s)$ of the Jacobian map $\s$. Analogous results are given for the Conjecture of Dixmier and the Poisson Conjecture. These results show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$ and the images of the Jacobian maps, of the endomorphisms of the Weyl algebra $A_n$ and of the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples.

A short direct algebraic (without reduction to prime characteristic) proof is given of the equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of the equivalence of the Jacobian, Poisson and Dixmier Conjectures).
\end{abstract}

\begin{altabstract}
Soit $P_n$ un polynôme à $n$ indéterminées sur un corps $K$ de caractéristique zéro. Un endomorphisme $\s\in \End_K(P_n)$ est appelé une \emph{application jacobienne} si son jacobien est un scalaire non nul. Chaque application jacobienne $\s$ est étendue en un endomorphisme $\s$ de l'algèbre de Weyl $A_n$.

La \emph{Conjecture Jacobienne} (CJ) affirme que chaque application jacobienne est un automorphisme. Clairement, la Conjecture Jacobienne est vraie si le module tordu (par $\s$) $\sP_n$ est cyclique pour toutes les applications jacobienne $\s$. Il est démontré que le module $A_n$ $\sP_n$ est cyclique pour toutes les applications jacobienne $\s$. De plus, le module $A_n$- $\sP_n$ est holonomique et, par conséquent, de longueur finie. Une borne supérieure explicite est trouvée pour la longueur du module $A_n$- $\sP_n$ en fonction du degré $\deg (\s)$ de l'application jacobienne $\s$. Des résultats analogues sont donnés pour la Conjecture de Dixmier et la Conjecture Poisson. Ces résultats montrent que la Conjecture Jacobienne, la Conjecture de Dixmier et la Conjecture Poisson sont des questions sur les modules holonomiques pour l'algèbre de Weyl $A_n$ et que les images des applications jacobienne, des endomorphismes de l'algèbre de Weyl $A_n$ et des endomorphismes de Poisson sont importantes au sens où des renforcements supplémentaires des résultats sur l'importance consisteraient soit à prouver les conjectures, soit à produire des contre-exemples.

Une démonstration directe et brève (sans réduction à la caractéristique première) est donnée de l'équivalence des Conjectures Jacobienne et Poisson (ceci donne une nouvelle démonstration brève de l'équivalence des Conjectures Jacobienne, Poisson et Dixmier).
\end{altabstract}

\keywords{\kwd{The Jacobian Conjecture}
\kwd{the Conjecture of Dixmier}
\kwd{the Weyl algebra}
\kwd{the holonomic module}
\kwd{the endomorphism algebra}
\kwd{the length}
\kwd{the multiplicity}}

\altkeywords{\kwd{La conjecture jacobienne}
\kwd{la conjecture de Dixmier}
\kwd{l'algèbre de Weyl}
\kwd{le module holonomique}
\kwd{l'algèbre des endomorphismes}
\kwd{la longueur}
\kwd{la multiplicité}}
\subjclass{14R15, 14R10, 13F20, 16S32, 14F10, 16D30, 16D60, 16P90}

\begin{document}
\maketitle

In this paper, $K$ is a field of characteristic zero and $K^\times := K\backslash \{ 0\}$, $P_n=K[x_1, \ldots, x_n]$ is a polynomial algebra in $n$ the variables, $\Der_K(P_n)$ is the set of all $K$-derivations of the polynomial algebra $P_n$. For a $K$-algebra $A$, the set $\End_K(A)$ is the monoid of $K$-algebra endomorphisms of $A$ and $\Aut_K(A)$ is the automorphism group of $A$.

\section{The Conjecture of Dixmier, holonomic \texorpdfstring{$A_n$}{An}-modules and finite length.}

For an endomorphism $\s\in \End_K (A)$ and an $A$-module $M$ we denote by $\sM$ the $A$-module $M$ \emph{twisted by $\s$}: $\sM=M$ (as a vector space) and
\[
a\cdot m:= \s (a)m \;\; \text{ for all } a\in A, \;\; m\in M.
\]

The ring of differential operators $A_n:=\CD (P_n)$ on the polynomial algebra $P_n$ is called the \emph{Weyl algebra}. The Weyl algebra $A_n$ is generated by the elements $x_1, \ldots, x_n, \der_1, \ldots, \der_n$ subject the defining relations: $[x_i, x_j]=0$, $[\der_i, \der_j]=$ and $[\der_i, x_j]=\delta_{ij}$ for all $i,j=1, \ldots, n$ where $\der_i:=\frac{\der}{\der x_i}$, $[a,b]:=ab-ba$, and $\delta_{ij}$ is the Kronecker delta. The Weyl algebra $A_n$ is a simple Noetherian domain of Gelfand--Kirillov dimension $\GK (A_n)=2n.$

\begin{enonce*}{Inequality of Bernstein}
For all nonzero finitely generated $A_n$-modules $M$,
\[
\GK (M)\geq n.
\]
\end{enonce*}

A finitely generated $A_n$-module $M$ is called \emph{holonomic} if $\GK (M)=n$. Each holonomic module has finite length and is a cyclic $A_n$-module, i.e. 1-generated. Each nonzero sub- or factor module of a holonomic module is holonomic.

\begin{enonce*}{Conjecture of Dixmier $\mathbf{DC}_{\bn}$}[\cite{Dix}]
$\End_K(A_n)=\Aut_K(A_n)$.
\end{enonce*}

The Weyl algebra $A_n$ is isomorphic to its opposite algebra $A_n^{op}$ via $A_n\to A_n^{op}$, $x_i\mapsto x_i$, $\der_i\mapsto \der_i$ for $i=1, \ldots, n$. So, the algebra $A_n\otimes A_n^{op}\simeq A_{2n}$ is isomorphic to the Weyl algebra $A_{2n}$. In particular, every $A_n$-\emph{bimodule} $N$ is a \emph{left} $A_{2n}$-module (${}_{A_n}N_{A_n}={}_{A_n\otimes A_n^{op}}N\simeq {}_{A_{2n}}N$). When we say that an $A_n$-\emph{bimodule} $N$ is \emph{holonomic} we mean that the corresponding left $A_{2n}$-module $N$ is holonomic. The Weyl algebra $A_n$ is a simple holonomic $A_n$-bimodule (since $\GK (A_n)=2n$ and $\GK (A_{2n})=4n$).


\begin{theo}[{\cite[Theorem~1.3]{Bav-RenQn}}]\label{MholsMhol}
If $M$ is a holonomic $A_n$-module and $\s\in \End_K(A_n)$ then the $A_n$-module $\sM$ is also a holonomic $A_n$-module (and as a result has finite length and is 1-generated over $A_n$).
\end{theo}

The Weyl algebra $A_n$ is a simple algebra. So, for each $\s\in \End_K(A_n)$, the image $\s (A_n)$ is isomorphic to the Weyl algebra $A_n$.
\begin{itemize}
\item \emph{The Conjecture of Dixmier is true if for every endomorphism $\s\in \End (A_n)$, the $\s(A_n)$-bimodule $A_n$ is simple.}
\end{itemize}

\begin{coro}[{\cite[Corollary~3.4]{Bav-RenQn}}]
For each algebra endomorphism $\s : A_n\to A_n$, the Weyl algebra $A_n$ is a holonomic $\s(A_n)$-bimodule, hence, of finite length and 1-generated.
\end{coro}

Each nonzero element $a\in A_n$ is a unique sum $a=\sum_{\alpha, \beta \in \N^n}\lambda_{\alpha\beta} x^\alpha \der^\beta$ for some scalars $\lambda_{\alpha\beta}\in K$ where $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ and $\der^\beta= \der_1^{\beta_1}\cdots \der_n^{\beta_n}$. The natural number
\[
\deg (a):=\max \{ |\alpha| +|\beta|\, | \, \lambda_{\alpha\beta}\neq 0\}
\]
is called the \emph{degree} of the element $a$. Then $\{ A_{n,i}\}_{i\geq 0}$ is a finite dimensional filtration of the Weyl algebra $A_n$ where $A_{n,i}:=\{ a\in A_n\, | \, \deg (a)\leq i\}$ and $\deg (0):=-\infty$ ($A_n=\bigcup_{i\geq 0}A_{n,i}$ and $A_{n,i}A_{n,j}\subseteq A_{n, i+j}$ for all $i,j\geq 0$).

Each endomorphism $\s \in \End_K(A_n)$ is uniquely determined by the elements
\[
x_1':=\s (x_1), \ldots, x_n':=\s (x_n), \der_i':=\s (\der_1), \ldots, \der_n':=\s (\der_n).
\]
The natural number $\deg (\s):=\max\{\deg (x_i'), \deg(\der_i')\, \ \, i=1,\ldots, n \}$ is called the \emph{degree} of $\s$.

\begin{theo}\label{A6Nov21}
Let $\s \in \End(A_n)$ and $d:=\deg (\s)$. Then $L_{\s(A_n)}(A_n)\leq d^{2n} $ where $L_{\s(A_n)}(M)$ is the length of a $\s(A_n)$-bimodule $M$.
\end{theo}

\begin{proof}
Since $\deg (x_i')\leq d$ and $\deg (\der_i')\leq d$,
\[
x_i'A_{n,ds}\subseteq A_{n, d(s+1)},\; A_{n,ds} x_i'\subseteq A_{n, d(s+1)},\; \der_i'A_{n,ds}\subseteq A_{n, d(s+1)}\quad\text{and}\quad A_{n,ds}\der_i'\subseteq A_{n, d(s+1)}\;
\]
for all $i=1, \ldots, n$ and $s\geq 0$. Therefore, $\{ A_{n, ds}\}_{s\geq 0}$ is a finite dimensional filtration of the $\s(A_n)$-bimodule $A_n$ such that
\[
\dim_K(A_{n,ds})=\binom{ds+2n}{2n}=\frac{1}{(2n)!} (ds+2n)(ds+2n-1)\cdots (ds+1)=\frac{d^{2n}}{(2n)!}s^{2n}+\cdots
\]
where three dots denote smaller terms. By~\cite[Lemma~8.5.9]{MR}, $L_{\s(A_n)}(A_n)\leq d^{2n} $.
\end{proof}

\section{The Jacobian Conjecture, holonomic \texorpdfstring{$A_n$}{An}-modules and 1-generation}

Each endomorphism $\s \in \End_K(P_n)$ is uniquely determined by the polynomials
\[
x_1':=\s (x_1), \ldots, x_n':=\s (x_n).
\]

The matrix of partial derivatives,
\[
\CJ (\s):=\frac{\der x'}{\der x}:=\Bigg(\frac{\der x_i'}{\der x_j}\Bigg), \quad\text{where}\quad \CJ (\s)_{ij}:=\frac{\der x_i'}{\der x_j},
\]
is called the \emph{Jacobian matrix} of $\s$. An endomorphism $\s\in \End (P_n)$ with $\det (\CJ (\s))\in K^\times$ is called a \emph{Jacobian map}.

\begin{enonce*}{Jacobian Conjecture $\mathbf{JC}_{\bn}$ }
Every Jacobian map is an automorphism.
\end{enonce*}

\begin{theo}[{\cite[Theorem~2.1]{BCW}}]\label{Thm2.1BCW}
A Jacobian map $\s\in \End (P_n)$ is an automorphism of $P_n$ if the $P_n$-module $\sP_n$ is finitely generated.
\end{theo}

\begin{itemize}
\item \emph{The Jacobian Conjecture is true iff the $P_n$-module $\sP_n$ is 1-generated for all Jacobian maps~$\s$}
\end{itemize}

Each Jacobian map $\s$ is extended to a (necessarily) monomorphism of the Weyl algebra $A_n$:
\begin{equation}\label{sAnext}
\s :A_n\to A_n, \;\; \der_i\mapsto \der_i', \;\; i=1, \ldots, n,
\end{equation}
where $\der_i'$ is a $K$-derivation of the polynomial algebra $P_n$ which is given by the rule:
\begin{equation}\label{derip}
\der_i'(p):=\frac{1}{\det \, \CJ (\s)}\CJ (\s (x_1), \ldots, \s (x_{i-1}), p, \s(x_{i+1}), \ldots, \s (x_n))\quad\text{for all}\quad p\in P_n.
\end{equation}

For an algebra $A$ and a non-empty subset $S$ of $A$, we define the \emph{centralizer} of $S$ in $A$ by $C_A(S):=\{ a\in A\, | \, as=sa$ for all $s\in S\}$. Let $\hP_n:= K\llbracket x_1, \ldots, x_n\rrbracket$ and $\hA_n:=\bigoplus_{\alpha\in \N^n}\hP_n\der^\alpha$. Proposition~\ref{B6Nov21} is a description of all extensions of a Jacobian map of $P_n$ to an endomorphism of the Weyl algebra~$A_n$.

\begin{prop}\label{B6Nov21}
Let $\s$ be a Jacobian map of $P_n$, $\s$ be its extension to an endomorphism of the Weyl algebra $A_n$ as in~\eqref{sAnext}, $x_1'=\s(x_1), \ldots, x_n'=\s (x_n)$ and $\der_1'=\s (\der_1), \ldots, \der_n'=\s (\der_n)$, see~\eqref{derip}.
\begin{enumerate}
\item \label{5.1} If $\s'$ is another extension of the Jacobian map $\s$ then $\s'(\der_i)=\der_i'+\der_i'(p)$, $i=1, \ldots, n$ where $p\in P_n$, and vice versa.
\item \label{5.2} An extension of the Jacobian map $\s$ of $P_n$ is unique if the images of the elements $\der_1, \ldots, \der_n$ are derivations of $P_n$. So, the extension $\s$ in~\eqref{derip} is such a unique extension, and $\Der_K(P_n)=\bigoplus_{i=1}^nP_n\der_i'$.
\item \label{5.3} $C_{A_n}(x_1', \ldots, x_n')=P_n$.
\item \label{5.4} Suppose that $x_i'=x_i+\cdots$ for $i=1, \ldots, n$ where the three dots denote higher terms. Then $\s\in \Aut_K(\hP_n)$ and every extension $\s'$ of the Jacobian map $\s$ to an endomorphism of the Weyl algebra $A_n$ belongs to $\Aut_K(\hA_n)$.
\end{enumerate}
\end{prop}

\begin{proof}\ 
\begin{proof}[(\ref{5.1})--(\ref{5.3})]
Up to an affine change of variables in the polynomial algebra $P_n$, we can assume that $\s (x_i)=x_i+\cdots$ for $i=1, \ldots, n$ where the three dots denote \emph{higher} terms. Since $\det (\CJ (\s))\in K^\times$, we have that $\Der_K(P_n)=\bigoplus_{i=1}^nP_n\der_i'$ (as $\der_i=\sum_{j=1}^n\frac{\der x_j'}{\der x_i}\der_j'$ for all $i=1, \ldots, n$), and so
\[
\hP_n= K\llbracket x_1', \ldots, x_n'\rrbracket\;\quad\text{and}\quad \s\in \Aut_K(\hP_n).
\]
If the elements $\s'(\der_i)$ are derivations of the polynomial algebra $P_n$ then
\[
\s' (\der_i)(x_j')=[\s' (\der_i), x_j']=[\s' (\der_i), \s'(x_j)]=\s' ([\der_i, x_j])=\s'(\delta_{ij})=\delta_{ij}\quad\text{for}\quad i,j=1, \ldots, n.
\]
Hence, $\s'(\der_i)=\frac{\der}{\der x_i'}$ for $i=1,\ldots, n$, and so $\s'(\der_i)=\der_i'$, see~\eqref{derip}.

In the general case,
\[
[\s'(\der_i)-\der_i',x_j']=[\s'(\der_i),\s' (x_j)]-[\der_i',x_j']=\delta_{ij}-\delta_{ij}=0,
\]
and so $d_i:=\s'(\der_i)-\der_i'\in C_{A_n}(x_1', \ldots, x_n')$. Clearly,
\[
P_n\subseteq C_{A_n}(x_1', \ldots, x_n')\subseteq C_{\hA_n}(x_1', \ldots, x_n')=\hat{P}_n,
\]
and so $ C_{A_n}(x_1', \ldots, x_n')=P_n$. Therefore, $d_i\in P_n\subseteq \hP_n= K\llbracket x_1', \ldots, x_n'\rrbracket$ for all $i=1, \ldots, n$. For all $i,j=1, \ldots, n$,
\[
0=\s'([\der_i, \der_j])=[\s'(\der_i), \s'(\der_j)]=[\der_i'+d_i, \der_j'+d_j]=\der_i'(d_j)-\der_j'(d_i).
\]
Therefore, there is an element $p\in K\llbracket x_1', \ldots, x_n'\rrbracket$ such that $d_i=\der_i'(p)$ for $i=1, \ldots, n$, by the Poincar\'{e} Lemma. Since all $d_j\in P_n$, we must have
\[
\der_i(p)=\sum_{j=1}^n\frac{\der x_j'}{\der x_i}\der_j'(p) =\sum_{j=1}^n\frac{\der x_j'}{\der x_i}d_j\in P_n.
\]
Hence, $p\in P_n$ since $K\llbracket x_1',\ldots, x_n'\rrbracket=\hP_n$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{5.4})]
It follows from statement \eqref{5.1}.
\end{proof}
\let\qed\relax
\end{proof}

\begin{theo}\label{6Nov21}
Let $\s \in \End(P_n)$ be Jacobian map and $d:=\deg (\s)$. Then
\begin{enumerate}
\item \label{6.1} The $A_n$-module $\sP_n$ is holonomic, hence of finite length and 1-generated as an $A_n$-module.
\item \label{6.2} $l_{A_n}(\sP_n)\leq m^n $ where $m:=\max \{d, (d-1)^{n-1}-1\}$ where $l_{A_n}(M)$ is the length of an $A_n$-module $M$.
\end{enumerate}
\end{theo}

\begin{proof}\ 
\begin{proof}[(\ref{6.1})]
The $A_n$-module $P_n$ is holonomic. By Theorem~\ref{MholsMhol}, the $A_n$-module $\sP_n$ is holonomic, hence of finite length and 1-generated as an $A_n$-module.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{6.2})]
Since $\deg (x_i')\leq d$,
\[
\der_i'=\sum_{j=1}^n\frac{\der x_j}{\der x_i'}\der_j=\sum_{j=1}^n\Big(\CJ (\s)^{-1}\Big)_{ij}\der_j, \quad\text{and}\quad \deg \Big(\CJ (\s)^{-1}\Big)_{ij}\leq (d-1)^{n-1},
\]
we have that
\[
x_i'P_{n,ms}\subseteq P_{n, m(s+1)},\quad\text{and}\quad \der_i'P_{n,ms}\subseteq P_{n, m(s+1)}\quad\text{for all}\quad i=1, \ldots, n\quad\text{and}\quad s\geq 0.
\]
Therefore, $\{ P_{n, ms}\}_{s\geq 0}$ is a finite dimensional filtration of the $A_n$-module ${}^{\s} P_n$ such that
\[
\dim_K(P_{n,ms})=\binom{ms+n}{n}=\frac{1}{n!} (ms+n)(ms+n-1)\cdots (ms+1)=\frac{m^n}{n!}s^n+\cdots
\]
where three dots denote smaller terms. By~\cite[Lemma~8.5.9]{MR}, $l_{A_n}(\sP_n)\leq m^n $.
\end{proof}
\let\qed\relax
\end{proof}

The Dixmier Conjecture implies the \emph{Jacobian Conjecture}, \cite[p.~297]{BCW}), and the inverse implication is also true, as shown by Tsuchimoto~\cite{Tsuchimoto-2005} and Belov--Kanel and Kontsevich~\cite{Belov-Konts-2007} (a short proof is given in~\cite{Bav-DCJC-2005}).


\section{Equivalence of the Jacobian and the Poisson Conjectures}

The Weyl algebra $A_n=\CD (P_n)=\bigcup_{i\geq 0}\CD (P_n)_i$ is a ring of differential operators on $P_n$ and hence admits the \emph{degree filtration} $\{ \CD (P_n)_i\}_{i\geq 0}$ where $\CD (P_n)_i=\bigoplus_{\{ \alpha\in \N^n\, | \, |\alpha |\leq i\}}P_n\der^\alpha$. The \emph{associated graded algebra}
\[
\gr(A_n):=\bigoplus_{i\geq 0}\gr(A_n)_i,
\]
where $\gr(A_n)_i:=\CD (P_n)_i/\CD (P_n)_{i-1}$ and $\CD (P_n)_{-1}:=0$, is a polynomial algebra $P_{2n}$ in $2n $ variables $x_1, \ldots, x_n, x_{n+1}, \ldots, x_{n+n}$ (where $ x_{n+i}:=\der_i+P_n$) that admits the canonical Poisson structure given by the rule:
\begin{equation}\label{grAnPois}
\{{\,\cdot\,, \cdot\,}\}: \gr(A_n)_i\otimes_K \gr(A_n)_j\to \gr(A_n)_{i+j-1}, \;\; (\oa, \ob)\mapsto \{ \oa, \ob\}:=[a,b]+\CD (P_n)_{i+j-2}
\end{equation}
where $\oa :=a+\CD (P_n)_{i-1}$ and $\ob :=b+\CD (P_n)_{j-1}$ (since $[\CD (P_n)_i,\CD (P_n)_j]\subseteq\CD (P_n)_{i+j-1}$ for all $i,j\geq 0$). Equivalently,
\begin{equation}\label{grAnPois1}
\{x_i, x_j\}=0,\;\; \{ x_{n+i}, x_{n+j}\}=0\quad\text{and}\quad \{ x_{n+i}, x_j\}= \delta_{ij}\quad\text{for all}\quad i,j=1, \ldots,n.
\end{equation}

\begin{enonce*}{Poisson Conjecture $\mathbf{PC}_{\boldsymbol{2n}}$}
$\End_{\mathrm{Pois}}(P_{2n})=\Aut_{\mathrm{Pois}}(P_{2n})$.
\end{enonce*}

The Poisson Conjecture and the Conjecture of Dixmier are equivalent (Adjamagbo and van den Essen \cite{Adjam-vdEs-2007}).

\begin{theo}\ 
\begin{enumerate}
\item \label{7.1} $\JC_{2n}\Rightarrow \PC_{2n}$.
\item \label{7.2} $\DC_{2n}\Rightarrow \PC_{2n}$.
\item \label{7.3} $\PC_{2n}\Rightarrow \JC_n$.
\item \label{7.4} The Jacobian Conjecture and the Poisson Conjecture are equivalent.
\item \label{7.5} The Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are equivalent.
\end{enumerate}
\end{theo}

\goodbreak
\begin{proof}\ 
\begin{proof}[(\ref{7.1})]
Given $\s\in \End_{\mathrm{Pois}}(P_{2n})$. Then $\det(\CJ (\s))\in \{ \pm 1\}$ (see the proof of Step 6 of~\cite[Theorem~3]{Bav-DCJC-2005} of the fact that $\JC_{2n}\Rightarrow \DC_n$): Notice that $\det(\{ x_i, x_j\})\in \{ \pm 1\}$ where $1\leq i,j\leq 2n$, and~so
\begin{align*}
\{ \pm 1\}&\ni \det(\{ x_i, x_j\})=\s(\det(\{ x_i, x_j \}))= \det(\s (\{ x_i, x_j\}))\\
&= \det(\{ \s(x_i), \s (x_j)\})=\det \bigl(\CJ^t(\s)\cdot (\{ x_i, x_j\})\cdot \CJ (\s)\bigr)\\
& =\det(\CJ(\s))^2\det(\{ x_i, x_j\}),
\end{align*}
and so $\det(\CJ (\s))\in \{ \pm 1\}$. By $\JC_{2n}$, $\s\in \Aut_K(P_{2n})$, and statement 1 follows.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{7.2})]
Given $\s\in \End_{\mathrm{Pois}}(P_{2n})$. The map
\begin{equation}\label{sPoisWA}
\s : A_{2n} \to A_{2n}, \;\; x_i\mapsto x_i':=
\s (x_i), \;\; \der_i\mapsto \der_i'({\,\cdot\,}):=
\begin{cases}
\{x_{n+i}', \cdot\,\} & \text{if }i=1,\ldots, n,\\
\{-x_{n-i}', \cdot\,\}& \text{if }i=n+1,\ldots, 2n,\\
\end{cases}
\end{equation}
is an algebra endomorphism of the Weyl algebra $A_{2n}$ where $\der_i'\in \Der_K(P_{2n})$. By $\DC_{2n}$, $\s\in \Aut_K(A_{2n})$, and so
\[
A_{2n}=\bigoplus_{\alpha, \beta \in \N^{2n}}Kx'^\alpha \der'^\beta.
\]
It follows that $\CD (P_{2n})_i=\bigoplus_{ \{ \alpha, \beta \in \N^{2n}\, | \, |\beta|\leq i\}}Kx'^\alpha \der'^\beta $ (use the defining relations in the new variables of $A_{2n}$). By the very definition, the automorphism $\s$ respects the degree filtration on $A_{2n}$. In particular, $\s (P_{2n})=P_{2n}$ since $\CD (P_{2n})_0=P_{2n}$, i.e. $\s \in \Aut_{\mathrm{Pois}}(P_{2n})$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{7.3})]
Given a Jacobian map $\s \in \End_K(P_n)$. Let $\s\in \End_K(A_n)$ be its extension given by~\eqref{sAnext}. By~\eqref{derip},
\[
\s (\CD (P_n)_i)\subseteq \CD (P_n)_i\quad\text{for all}\quad i\geq 0,
\]
{i.e.} the endomorphism $\s$ of the Weyl algebra $A_n$ respects the degree filtration and so the associated graded map
\begin{equation}\label{grsPois}
\gr(\s): \gr(A_n)\to \gr(A_n), \;\; a+\CD (P_n)_{i-1}\mapsto \s (a)+\CD (P_n)_{i-1}
\end{equation}
respects the Poisson structure, i.e. $\gr(\s) \in \End_{\mathrm{Pois}}(\gr(A_n))$. By $\PC_{2n}$, $\gr(\s) \in \Aut_{\mathrm{Pois}}(\gr(A_n))$, hence $\s\in \Aut_K(P_n)$ since the automorphism $\gr(\s) \in \Aut_{\mathrm{Pois}}(\gr(A_n))$ is a \emph{graded} automorphism and $\CD (P_n)_0=P_n$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{7.4})]
It follows from statements \eqref{7.1} and \eqref{7.3}.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{7.5})]
It follows from statement~\eqref{7.4} and the equivalence $\JC_{2n}\Longleftrightarrow \DC_n$.
\end{proof}
\let\qed\relax
\end{proof}

By~\eqref{sPoisWA},
\begin{itemize}
\item \emph{$\PC_{2n}$ is true iff the $A_{2n}$-module $\sP_{2n}$ is simple for all $\s\in \End_{\mathrm{Pois}}(P_{2n})$.}
\end{itemize}

\begin{theo}\label{16Nov21}
Let $\s \in \End_{\mathrm{Pois}}(P_{2n})$ and $d:=\deg (\s)$. Then
\begin{enumerate}
\item \label{8.1} The $A_{2n}$-module $\sP_{2n}$ is holonomic, hence of finite length and 1-generated as an $A_{2n}$-module.
\item \label{8.2} $l_{A_{2n}}(\sP_{2n})\leq d^{2n} $.
\end{enumerate}
\end{theo}

\begin{proof}\ 
\begin{proof}[(\ref{8.1})]
The $A_{2n}$-module $P_{2n}$ is holonomic. By Theorem~\ref{MholsMhol}, the $A_{2n}$-module $\sP_{2n}$ is holonomic, hence of finite length and 1-generated as an $A_{2n}$-module.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{8.2})]
Since $\deg (x_i')\leq d$ and $\deg (\der_i')\leq d$ (see~\eqref{sPoisWA}),
\[
x_i'P_{2n,ds}\subseteq P_{2n, d(s+1)},\quad\text{and}\quad \der_i'P_{2n,ds}\subseteq P_{2n, d(s+1)}\quad\text{for all}\quad i=1, \ldots, 2n\quad\text{and}\quad s\geq 0.
\]
Therefore, $\{ P_{2n, ds}\}_{s\geq 0}$ is a finite dimensional filtration of the $A_{2n}$-module ${}^{\s} P_{2n}$ such that
\[
\dim_K(P_{2n,ds})=\binom{ds+2n}{2n}=\frac{1}{(2n)!} (ds+2n)(ds+2n-1)\cdots (ds+1)=\frac{d^{2n}}{(2n)!}s^{2n}+\cdots
\]
where three dots denote smaller terms. By~\cite[Lemma~8.5.9]{MR}, $l_{A_{2n}}(\sP_{2n})\leq d^{2n}$.
\end{proof}
\let\qed\relax
\end{proof}


\section{An analogue of the Conjecture of Dixmier for the algebras \texorpdfstring{$\mI_n$}{In} of integro-differential operators}

Let $\mI_n:=K\langle x_1, \ldots, x_n, \der_1, \ldots,\der_n, \int_1, \ldots, \int_n\rangle $ be the algebra of polynomial integro-differential operators where $\int_i: P_n\to P_n$, $ p\mapsto \int p \, \dd x_i$, i.e. $\int_i x^\alpha = (\alpha_i+1)^{-1}x_ix^\alpha$ for all $\alpha \in \N^n$, \cite{intdifaut}.

\begin{conj*}[\cite{IntDifDixConj}]
$\End_K(\mI_n)=\Aut_K(\mI_n)$.
\end{conj*}

\begin{theo}[{\cite[Theorem~1.1]{IntDifDixConj}}]
$\End_K(\mI_1)=\Aut_K(\mI_1)$.
\end{theo}

\section{An analogue of the Jacobian Conjecture and the Conjecture of Dixmier for the algebras \texorpdfstring{$A_{n,m}:=A_n\otimes P_m$}{An,m :) An otimes Pm}} 

The centre of the algebra $A_{n,m}$ is $P_m$. Hence, for all $\s\in \Aut_K(A_{n,m})$, $\s (P_m)=P_m$.

\begin{enonce*}{Conjecture $\mathbf{JD}_{\boldsymbol{n,m}}$}[\cite{Bav-InvForWeylAlgPol-2007}]
Every endomorphism $\s :A_n\otimes P_m\to A_n\otimes P_m$ such that $\s (P_m)\subseteq P_m$ and $\det \bigg(\frac{\der \s (x_i)}{\der x_j}\bigg)\in K^*$ is an automorphism.
\end{enonce*}

\begin{theo}[{\cite[Theorem~5.8, Proposition~5.9]{Bav-InvForWeylAlgPol-2007}}]
$\JD_{n,m} \Longleftrightarrow \JC_m +\DC_n$.
\end{theo}

\section*{Acknowledgements}

The author would like to thank H. Bass and D. Wright for the recent discussions on the Jacobian Conjecture (November 2021) and the Royal Society for support.


\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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