Difference between revisions
dixmier-a2
Status and infobox
- status:
—→ open - kind:
—→ conjecture - statement:
—→ Over a field of characteristic $0$, every algebra endomorphism of the second Weyl algebra $A_2$ is an automorphism. - short:
—→ DC₂ — endomorphisms of A₂; open, and a proof would settle JC₂
Relations
- + special case of dixmier-conjecture: 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: The Vaserstein–Kac pullback at turns a non-invertible Keller map of the plane into a non-surjective endomorphism of , so DC₂ implies JC₂.
Source +15 −0
+---
+title: DC₂: the Dixmier conjecture for the second Weyl algebra
+short: DC₂ — endomorphisms of A₂; open, and a proof would settle JC₂
+symbol: DC₂
+kind: conjecture
+status: open
+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 [[special-case-of::Statement:dixmier-conjecture|DC₂ is a special case of the Dixmier conjecture in rank n]]. 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 [[implies::Statement:jc-plane|DC₂ implies JC₂]]. A proof of DC₂ would settle the plane case. A counterexample to JC₂ would refute DC₂.<ref>A. Belov-Kanel, M. Kontsevich, [The Jacobian conjecture is stably equivalent to the Dixmier conjecture](https://arxiv.org/abs/math/0512171), §1.</ref>
+
+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.<ref>C. D. Long, [An Explicit Counterexample to the Rank-Two Poisson Conjecture](https://arxiv.org/abs/2608.23777), arXiv:2608.23777 (22 July 2026).</ref> See [[Statement:poisson-conjecture|PCₙ]].
+
+<references />
+
Revision 247 as rendered
DC₂ is the rank-two case of the family, so DC₂ is a special case of the Dixmier conjecture in rank n. 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 turns a non-invertible Keller map of the plane into a non-surjective endomorphism of , so DC₂ implies JC₂. 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.
- ↑ A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, §1.
- ↑ C. D. Long, An Explicit Counterexample to the Rank-Two Poisson Conjecture, arXiv:2608.23777 (22 July 2026).