The Poisson conjecture in rank n

A Poisson endomorphism of $P_n$ is a polynomial map of $k^{2n}$ that preserves the standard symplectic form, so its Jacobian determinant is 1. The Poisson conjecture says every such map is invertible. It is the commutative shadow of the Dixmier conjecture: reducing the Weyl algebra modulo $p$ gives exactly this Poisson structure on its center. Adjamagbo and van den Essen introduced it to split the stable equivalence of the Jacobian and Dixmier conjectures into steps.^[1]

The chain

Their United Conjectures Theorem, degree-refined and valid in any characteristic, reads $\mathrm{JC}_{2n}\Rightarrow\mathrm{PC}_n\Rightarrow\mathrm{DC}_n\Rightarrow\mathrm{JC}_n$.^[1]

• A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2nn→2n.
• Reduction modulo $p$ is where the content lies: PCₙ implies the Dixmier conjecture in the same rankn→n.
• A shorter route skips the Weyl algebra. The cotangent lift $(x,\xi)\mapsto(F(x),JF(x)^{-T}\xi)$ of a Keller map is symplectic, and it is injective only if $F$ is, so PCₙ implies JCₙ directlyn→n.

At $n=1$ a Poisson endomorphism of $k[x,y]$ is a pair $(f,g)$ with $\{f,g\}=1$, which is a Jacobian pair. So PC₁ is JC₂ restated.

Status

• Rank 3 and up are false. The cotangent lift of Alpöge's map is an explicit symplectic, non-injective polynomial map of ℂ⁶.
• Rank 2 is claimed false. Long gives explicit $R,T,D,S\in\mathbb Q[x,q,p,z]$ with $R=x(2-3xq)$, in his convention $\{p,x\}=\{z,q\}=1$. The map $(x,q,p,z)\mapsto(R,T,D,S)$ is a non-automorphic Poisson endomorphism whose fiber over $(0,\tfrac18,0,0)$ is exactly three rational points: $(0,0,\tfrac1{24},-\tfrac18)$, $(1,\tfrac23,\tfrac{247}{96},-\tfrac{89}{64})$ and $(-1,-\tfrac23,\tfrac{247}{96},-\tfrac{89}{64})$.^[2] The map is built from Alpöge's, and it is not a cotangent lift: a lift from dimension 2 would need a counterexample to JC₂. The atlas has not re-checked it, because the formula for $D$ involves a degree-23 correction the atlas has not transcribed. The cell is drawn as a claim.
• Rank 1 is JC₂, which is open.

Because the implication runs PCₙ ⇒ DCₙ, a false PC₂ says nothing about DC₂.

1. ↑ a b K. Adjamagbo, A. van den Essen, On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic, arXiv:math/0608009 (2006), Theorem 7; published as Acta Math. Vietnam. 32 (2007) 205–214.
2. ↑ C. D. Long, An Explicit Counterexample to the Rank-Two Poisson Conjecture, arXiv:2608.23777 (submitted 22 July 2026, 17 pp.). Unrefereed.