The Dixmier conjecture in rank n

The Weyl algebra $A_n$ is the algebra of polynomial differential operators in $n$ variables. It is simple, so every endomorphism is injective. The Dixmier conjecture says every endomorphism is also surjective. Dixmier asked the question for $A_1$ in 1968, as the first of six problems about the first Weyl algebra.^[1] The rank-$n$ version is the noncommutative twin of the Jacobian conjecture.

Two implications, one in each direction

DCₙ implies JCₙ (Vaserstein and Kac, early 1980s, recorded by Bass–Connell–Wright). A Keller map $F$ gives the endomorphism $x_i\mapsto F_i$, $\partial_i\mapsto D_i=\sum_j\big((JF)^{-1}\big)_{ji}\,\partial_j$. Here $D_i(F_k)=\delta_{ik}$ and the $D_i$ commute. For étale $F$ the centralizer of $K[F]$ in $A_n$ is $K[x]$, by a principal-symbol argument. So if the endomorphism were onto, it would be an automorphism and $K[F]=K[x]$. In short, DCₙ implies JCₙ in the same dimensionn→n, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.^[2]

JC₂ₙ implies DCₙ (Tsuchimoto 2005; Belov-Kanel and Kontsevich 2007). Reduce modulo a large prime $p$. The center of $A_n$ becomes a polynomial ring in $2n$ variables with a Poisson bracket, and an endomorphism of $A_n$ induces a symplectic map of that space, whose Jacobian is 1. So DCₙ reduces to the Jacobian conjecture in twice the dimensionn→2n: a proof of JC₂ₙ would prove DCₙ.^[2] See the stable equivalence.

Together the two implications make the conjectures equivalent over all dimensions at once, though at no fixed $n$.

After July 2026

Rank 3 and up are false, with an explicit witness. Alpöge's map has integer coefficients and constant Jacobian $-2$, so $(JF)^{-1}$ has polynomial entries. The Vaserstein–Kac pullback is then a concrete endomorphism of $A_3$ over $\mathbb Q$, and its image misses some $x_i$. Tensoring with the identity carries it to every $A_n$ with $n\ge3$. Long's rank-two Poisson preprint also writes down an explicit non-automorphic endomorphism of $A_4$, in its Appendix B.^[3] The Fong–Fable ledger listed an explicit non-surjective endomorphism of $A_3$ as open question O4. The classical pullback already provides one. Whether quantizing the ledger's master equation gives a different or simpler one is not addressed.^[4]

Ranks 1 and 2 are open. Falsity passes up the ranks but never down, and the only implication still feeding the low ranks is JC₂ ⇒ DC₁, whose premise is open. What is left is the chain DC₂ ⇒ JC₂ ⇒ DC₁. Zheglov has claimed a proof of DC₁ (arXiv, 78 pages, unrefereed).^[5]

Evidence

V3 for the refutation of ranks $n\ge3$. The input counterexample is kernel-checked, and the pullback satisfies the Weyl relations by a finite identity. The step from “pullback of a non-invertible map” to “not surjective” is the classical centralizer argument, a written proof that has not been formalized.

1. ↑ J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209–242. The “first of six problems” reading follows J. A. Guccione, J. J. Guccione, C. Valqui, The Dixmier conjecture and the shape of possible counterexamples, J. Algebra 399 (2014).
2. ↑ a b A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Math. J. 7 (2007) 209–218, §1: “It is well-known that DC_n implies JC_n … The argument is very easy.”
3. ↑ C. D. Long, An Explicit Counterexample to the Rank-Two Poisson Conjecture, arXiv:2608.23777 (22 July 2026), Appendix B. Unrefereed.
4. ↑ D. Fong and Claude Fable 5, claims ledger, O4.
5. ↑ A. Zheglov, The Conjecture of Dixmier for the first Weyl algebra is true, arXiv:2410.06959 (v1 9 October 2024; v5 19 January 2026). Unrefereed.