The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture

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The Jacobian conjecture ${\rm JC}_n$ in dimension $n\geq 1$ asserts that for any field $\Bbbk$ of characteristic zero, any polynomial endomorphism $\phi$ of the $n$-dimensional affine space $\Bbb A^n_{\Bbbk}={\rm Spec}\,\Bbbk[x_1,\dots,x_n]$ over $\Bbbk$ with Jacobian 1, i.e. $$ \det(\partial\phi^*(x_i)/\partial x_j)_{1\leq i,j\leq n}=1, $$ is an automorphism. Equivalently, one can say that $\phi$ preserves the standard top-degree differential form $dx_1\wedge\dots\wedge dx_n\in\Omega^n(\Bbb A^n_{\Bbbk})$. "References pertaining to this well-known problem and related questions can be found in [A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000; MR1790619; H. Bass, E. H. Connell and D. Wright, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330; MR0663785]. "By the Lefschetz principle it is sufficient to consider the case $\Bbbk=\Bbb C$. Obviously, ${\rm JC}_n$ implies ${\rm JC}_m$ if $n > m$. We denote by ${\rm JC}_{\infty}$ the stable Jacobian conjecture, i.e., the conjunction of conjectures ${\rm JC}_n$ for all finite $n$. The conjecture ${\rm JC}_n$ is obviously true in the case $n=1$, and it is open for $n\ge 2$. "The Dixmier conjecture ${\rm DC}_n$ for integers $n\geq 1$ [see J. Dixmier, Bull. Soc. Math. France 96 (1968), 209–242; MR0242897] asserts that for any field $\Bbbk$ of characteristic zero, any endomorphism of the $n$-th Weyl algebra $A_{n,\Bbbk}$ over $\Bbbk$ is an automorphism. "Here $A_{n,\Bbbk}$ is the associative unital algebra over $\Bbbk$ with $2n$ generators $y_1,\dots,y_{2n}$ and relations $$ [y_i, y_j]=\omega_{ij}, $$ where $(\omega_{ij})_{1\leq i,j\leq 2n}$ is the following standard $2n\times 2n$ skew-symmetric matrix: $$ \omega_{ij}=\delta_{i,j+n}-\delta_{i+n,j}. $$ The algebra $A_{n,\Bbbk}$ coincides with the algebra $D(\Bbb A^n_{\Bbbk})$ of polynomial differential operators on $\Bbb A^n_{\Bbbk}$. For any $i$, $1\leq i\leq n$, element $y_i$ acts as the multiplication operator by the variable $x_i$, and element $y_{n+i}$ acts by the differentiation $\partial/\partial x_i$. Again, it is sufficient to consider the case $\Bbbk=\Bbb C$. The conjecture ${\rm DC}_n$ implies ${\rm DC}_m$ for $n>m$, and we can consider the stable Dixmier conjecture ${\rm DC}_{\infty}$. The conjecture ${\rm DC}_n$ is open for any $n\geq 1$. "It is well known that ${\rm DC}_n$ implies ${\rm JC}_n$ (in particular ${\rm DC}_{\infty}$ implies ${\rm JC}_{\infty}$) [see A. van den Essen, op. cit.; H. Bass, E. H. Connell and D. Wright, op. cit.]. "Our result is an opposite implication. Namely, we prove the following: Theorem 1. Conjecture ${\rm JC}_{2n}$ implies ${\rm DC}_n$. "In particular, we obtain that the stable conjectures ${\rm JC}_{\infty}$ and ${\rm DC}_{\infty}$ are equivalent. "Van den Essen [op. cit. (Theorem 10.4.2)] proved a weaker result: the conjecture ${\rm JC}_{2n}$ implies the invertibility of any endomorphism of $A_{n,\Bbbk}=D(\Bbb A^n_{\Bbbk})$ preserving the filtration by the degrees of differential operators

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