Dixmier conjecture

In algebra, the Dixmier conjecture, stated by Jacques Dixmier in 1968,¹ originally asked whether any endomorphism of the first Weyl algebra ${\displaystyle A_{1}}$ over a field of characteristic zero is an automorphism. The analogous statement for the ${\displaystyle n}$-th Weyl algebra ${\displaystyle A_{n}}$ was recorded in 1982 by Bass, Connell, and Wright, who attributed it to communications from Leonid Vaserstein and Victor Kac,² and was later referred to as the "generalized Dixmier conjecture"³⁴. For ${\displaystyle n\geq 3}$, it was disproved in 2026 as a consequence of a counterexample to the ${\displaystyle n}$-dimensional Jacobian conjecture.

Tsuchimoto in 2005,⁴ and independently Belov-Kanel and Kontsevich in 2007,⁵ showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture: the Dixmier conjecture for the n-th Weyl algebra ${\displaystyle A_{n}}$ implies the Jacobian conjecture for polynomial maps in n variables, while conversely the Jacobian conjecture in ${\displaystyle 2n}$ variables implies the Dixmier conjecture for ${\displaystyle A_{n}}$.⁵ In July 2026, a counterexample to the Jacobian conjecture in three variables was found,⁶ which by the first of these implications shows that the Dixmier conjecture is false for ${\displaystyle A_{n}}$ for all ${\displaystyle n\geq 3}$.

The conjecture remains open for the first and second Weyl algebras, since the Jacobian conjecture is still open in two variables. The case of the first Weyl algebra ${\displaystyle A_{1}}$ was the problem originally posed by Dixmier; a proposed proof of this case was announced by Alexander Zheglov in 2024.⁷