# The Poisson conjecture in rank n

- kind: family
- status: refuted (evidence tier V3)
- year: 2006
- authors: K. Adjamagbo and A. van den Essen (2006; Acta Math. Vietnam. 2007)
- in one line: PCₙ — Poisson endomorphisms of k[x₁…xₙ, y₁…yₙ] are automorphisms

**Statement.** Let $P_n=k[x_1,\dots,x_n,y_1,\dots,y_n]$, $\operatorname{char}k=0$, with $\{x_i,y_j\}=\delta_{ij}$ and $\{x_i,x_j\}=\{y_i,y_j\}=0$. $\mathrm{PC}_n$: every Poisson endomorphism of $P_n$ is an automorphism.

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 2n](/wiki/Statement:jc-n.md).
- Reduction modulo $p$ is where the content lies: [PCₙ implies the Dixmier conjecture in the same rank](/wiki/Statement:dixmier-conjecture.md).
- 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 directly](/wiki/Statement:jc-n.md).

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

| | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
|---|---|---|---|---|---|---|
| PCₙ | open | claimed false (V4; open) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |
| 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) |

## 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.

## Relations

- reduces to [jc-n](/wiki/Statement:jc-n.md) — A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.
- implies [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md) — Reduction modulo $p$ is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.
- implies [jc-n](/wiki/Statement:jc-n.md) — 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 directly.

## References

1. K. Adjamagbo, A. van den Essen, [On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic](https://arxiv.org/abs/math/0608009), 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](https://arxiv.org/abs/2608.23777), arXiv:2608.23777 (submitted 22 July 2026, 17 pp.). Unrefereed.

## Relations to this page

- related to [tsuchimoto-bkk](/wiki/Statement:tsuchimoto-bkk.md) — Adjamagbo and van den Essen gave a third proof, purely algebraic, through the Poisson conjecture, which isolates the symplectic step as its own statement.

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