# DC₂: the Dixmier conjecture for the second Weyl algebra

- kind: conjecture
- status: open
- in one line: DC₂ — endomorphisms of A₂; open, and a proof would settle JC₂

**Statement.** Over a field of characteristic $0$, every algebra endomorphism of the second Weyl algebra $A_2$ is an automorphism.

DC₂ is the rank-two case of the family, so [DC₂ is a special case of the Dixmier conjecture in rank n](/wiki/Statement:dixmier-conjecture.md). It survived July 2026 for the same reason DC₁ did: nothing false feeds into it.

It sits above the plane Jacobian conjecture. The Vaserstein–Kac pullback at $n=2$ turns a non-invertible Keller map of the plane into a non-surjective endomorphism of $A_2$, so [DC₂ implies JC₂](/wiki/Statement:jc-plane.md). A proof of DC₂ would settle the plane case. A counterexample to JC₂ would refute DC₂.[1]

The route through the Poisson conjecture does not reach it. The chain JC₄ ⇒ PC₂ ⇒ DC₂ needs PC₂, and Long claims an explicit counterexample to PC₂ (unrefereed). If that claim holds, the implication PC₂ ⇒ DC₂ says nothing, and DC₂ stays open.[2] See [PCₙ](/wiki/Statement:poisson-conjecture.md).

## Relations

- special case of [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md) — DC₂ is the rank-two case of the family, so DC₂ is a special case of the Dixmier conjecture in rank n.
- implies [jc-plane](/wiki/Statement:jc-plane.md) — The Vaserstein–Kac pullback at $n=2$ turns a non-invertible Keller map of the plane into a non-surjective endomorphism of $A_2$, so DC₂ implies JC₂.

## References

1. A. Belov-Kanel, M. Kontsevich, [The Jacobian conjecture is stably equivalent to the Dixmier conjecture](https://arxiv.org/abs/math/0512171), §1.
2. C. D. Long, [An Explicit Counterexample to the Rank-Two Poisson Conjecture](https://arxiv.org/abs/2608.23777), arXiv:2608.23777 (22 July 2026).

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