DC₁: Dixmier's problem for the first Weyl algebra

This is the question Dixmier asked, the first of the six problems in his 1968 paper on the Weyl algebras. It is the rank-one case of the family, so DC₁ is a special case of the Dixmier conjecture in rank n. The ranks from three up are now false, but falsity passes up the ranks, never down, and this case stands.

It is controlled by the plane. By Tsuchimoto and Belov-Kanel–Kontsevich, a proof of the two-dimensional Jacobian conjecture would prove DC₁ (the relation is recorded on JC₂). A counterexample to DC₁ would therefore refute JC₂.^[1]

A claimed proof exists. Zheglov's preprint (78 pages, fifth version January 2026) proves DC₁ over any field of characteristic zero. It has no journal reference, and the atlas records it as a claim (V4), not a settlement.^[2] Guccione, Guccione and Valqui describe the shape any counterexample would have to take.^[3]

1. ↑ A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Math. J. 7 (2007) 209–218.
2. ↑ A. Zheglov, The Conjecture of Dixmier for the first Weyl algebra is true, arXiv:2410.06959 (v5 19 January 2026).
3. ↑ J. A. Guccione, J. J. Guccione, C. Valqui, The Dixmier conjecture and the shape of possible counterexamples, J. Algebra 399 (2014) 581–633.