The Jacobian Conjecture\(_{2n}\) implies the Dixmier Problem\(_n\) (Q1980731)

The Dixmeier Problem \(DP_n\) predicts that every \(\mathbb C\)-algebra endomorphism of the Weyl algebra \(A_n (\mathbb C)\) is an algebra automorphism. Here the author describes some ideas regarding the Dixmeier Conjecture and its generalizations and analogues. In particular, he gives a short proof that the \(2n\)-variable Jacobian conjecture implies \(DP_n\) using the inversion formula for automorphisms of the Weyl algebras with polynomial coefficients and his bound on its degree [\textit{V. V. Bavula}, J. Pure Appl. Algebra 210, No. 1, 147--159 (2007; Zbl 1128.13017)]. This result was proved independently by \textit{Y. Tsuchimoto} [Osaka J. Math. 42, No. 2, 435--452 (2005; Zbl 1105.16024)] and \textit{A. Belov-Kanel} and \textit{M. Kontsevich} [Mosc. Math. J. 7, No. 2, 209--218 (2007; Zbl 1128.16014)], where \(JC_{2n}\) is the Jacobian conjecture in \(2n\) variables. \textit{P. K. Adjamagbo} and \textit{A. van den Essen} also proved this going through the Poisson conjecture [Acta Math. Vietnam. 32, No. 2--3, 205--214 (2007; Zbl 1137.14046)]. For the entire collection see [Zbl 1467.16001].