# DC₁: Dixmier's problem for the first Weyl algebra

- kind: conjecture
- status: open
- year: 1968
- authors: J. Dixmier
- in one line: DC₁ — is every endomorphism of A₁ an automorphism? Open

**Statement.** Over a field of characteristic $0$, every algebra endomorphism of $A_1=K\langle x,\partial\rangle$, $[\partial,x]=1$, is an automorphism.

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](/wiki/Statement:dixmier-conjecture.md). 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](/wiki/Statement:jc-plane.md)). 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]

## Relations

- special case of [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md) — It is the rank-one case of the family, so DC₁ is a special case of the Dixmier conjecture in rank n.

## References

1. A. Belov-Kanel, M. Kontsevich, [The Jacobian conjecture is stably equivalent to the Dixmier conjecture](https://arxiv.org/abs/math/0512171), Moscow Math. J. 7 (2007) 209–218.
2. A. Zheglov, [The Conjecture of Dixmier for the first Weyl algebra is true](https://arxiv.org/abs/2410.06959), 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](https://arxiv.org/abs/1111.6100), J. Algebra 399 (2014) 581–633.

## Relations to this page

- implied by [jc-plane](/wiki/Statement:jc-plane.md) — By Tsuchimoto and Belov-Kanel–Kontsevich, JC₂ implies Dixmier's original problem DC₁ for the first Weyl algebra. In the other direction, DC₂ implies JC₂, and so does the four-variable Hessian conjecture HC₄. The rank-one Poisson conjecture is JC₂ restated: a Poisson endomorphism of $k[x,y]$ is exactly a pair with Jacobian 1 (see PCₙ).

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