# The Dixmier conjecture in rank n

- kind: family
- status: refuted (evidence tier V3)
- year: 1968
- authors: J. Dixmier (the rank-one question, 1968); the rank-n form recorded by Bass, Connell and Wright (1982), credited to Vaserstein and Kac
- in one line: DCₙ — every endomorphism of the Weyl algebra Aₙ is an automorphism

**Statement.** Let $K$ have characteristic $0$ and let $A_n(K)=K\langle x_1,\dots,x_n,\partial_1,\dots,\partial_n\rangle$ with $[\partial_i,x_j]=\delta_{ij}$. $\mathrm{DC}_n$: every $K$-algebra endomorphism of $A_n(K)$ is an automorphism.

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 dimension](/wiki/Statement:jc-n.md), 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 dimension](/wiki/Statement:jc-n.md): a proof of JC₂ₙ would prove DCₙ.[2] See [the stable equivalence](/wiki/Statement:tsuchimoto-bkk.md).

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

## After July 2026

| | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
|---|---|---|---|---|---|---|
| DCₙ | claimed true (V4; open) | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |
| JCₙ | proved | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |

**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](/wiki/Statement:dixmier-a2.md) ⇒ [JC₂](/wiki/Statement:jc-plane.md) ⇒ [DC₁](/wiki/Statement:dixmier-a1.md). 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.

## Relations

- implies [jc-n](/wiki/Statement:jc-n.md) — In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra. [2]
- reduces to [jc-n](/wiki/Statement:jc-n.md) — So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence. [2]

## References

1. J. Dixmier, [Sur les algèbres de Weyl](http://www.numdam.org/item/BSMF_1968__96__209_0/), 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](https://arxiv.org/abs/1111.6100), J. Algebra 399 (2014).
2. 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, §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](https://arxiv.org/abs/2608.23777), arXiv:2608.23777 (22 July 2026), Appendix B. Unrefereed.
4. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), O4.
5. A. Zheglov, [The Conjecture of Dixmier for the first Weyl algebra is true](https://arxiv.org/abs/2410.06959), arXiv:2410.06959 (v1 9 October 2024; v5 19 January 2026). Unrefereed.

## Relations to this page

- generalizes [dixmier-a1](/wiki/Statement:dixmier-a1.md) — It is the rank-one case of the family, so DC₁ is a special case of the Dixmier conjecture in rank n.
- generalizes [dixmier-a2](/wiki/Statement:dixmier-a2.md) — DC₂ is the rank-two case of the family, so DC₂ is a special case of the Dixmier conjecture in rank n.
- implied by [poisson-conjecture](/wiki/Statement:poisson-conjecture.md) — Reduction modulo $p$ is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.
- related to [tsuchimoto-bkk](/wiki/Statement:tsuchimoto-bkk.md) — The theorem supplies the implication recorded on the Dixmier family as “DCₙ reduces to JC₂ₙ”, which with DCₙ ⇒ JCₙ sandwiches DCₙ between JCₙ and JC₂ₙ.

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