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poisson-conjecture

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Revision 275 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__poisson-conjecture.md).”

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+---
+title: The Poisson conjecture in rank n
+short: PCₙ — Poisson endomorphisms of k[x₁…xₙ, y₁…yₙ] are automorphisms
+symbol: PC
+kind: family
+status: refuted
+tier: V3
+year: 2006
+authors: K. Adjamagbo and A. van den Essen (2006; Acta Math. Vietnam. 2007)
+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.
+index: n
+cells: 1 = open | 2 = refuted V4 witnessed | 3 = refuted V3 witnessed
+stable: yes
+---
+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.<ref name="ave">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.</ref>
 
+## The chain
+ 
+Their United Conjectures Theorem, degree-refined and valid in any characteristic, reads $\mathrm{JC}_{2n}\Rightarrow\mathrm{PC}_n\Rightarrow\mathrm{DC}_n\Rightarrow\mathrm{JC}_n$.<ref name="ave"/>
+ 
+- A symplectic map has Jacobian 1, so [[reduces-to::Statement:jc-n|PCₙ reduces to the Jacobian conjecture in dimension 2n]]{{at|n→2n}}.
+- Reduction modulo $p$ is where the content lies: [[implies::Statement:dixmier-conjecture|PCₙ implies the Dixmier conjecture in the same rank]]{{at|n→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 [[implies::Statement:jc-n|PCₙ implies JCₙ directly]]{{at|n→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 [[Statement:jc-plane|JC₂]] restated.
+ 
+{{ladder|Statement:poisson-conjecture|Statement:dixmier-conjecture|Statement:jc-n|n=1..6}}
+ 
+## 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 $(-1,-\tfrac23,\tfrac{247}{96},-\tfrac{89}{64})$.<ref name="long">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.</ref> 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.
+ 
+<references />
+ 

Revision 275 as rendered

A Poisson endomorphism of PnP_n is a polynomial map of k2nk^{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 pp 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 JC2n⇒PCn⇒DCn⇒JCn\mathrm{JC}_{2n}\Rightarrow\mathrm{PC}_n\Rightarrow\mathrm{DC}_n\Rightarrow\mathrm{JC}_n.[1]

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

n = 1n = 2n = 3n = 4n = 5n = 6…padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.n→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: 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.n→n: 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.n→n: 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.n→n: 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.n→n: 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.n→n: 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.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.PCnPoisson conjecturePC₁: open: nothing settles itPC₁openPC₂: claimed false (V4, not yet adjudicated or checked; still open)PC₂claimedPC₃: refuted (declared, tier V3, explicit witness)PC₃PC₄: false by implication, tier V3, explicit witness — paddingPC₄PC₅: false by implication, tier V3, explicit witness — padding → paddingPC₅PC₆: false by implication, tier V3, explicit witness — padding → padding → paddingPC₆PCₙ, n > 6: false by implication, tier V3, explicit witnessPCDCnDixmier conjectureDC₁: claimed true (V4, not yet adjudicated or checked; still open)DC₁claimedDC₂: open: nothing settles itDC₂openDC₃: refuted (declared, tier V3, explicit witness)DC₃DC₄: false by implication, tier V3, explicit witness — paddingDC₄DC₅: false by implication, tier V3, explicit witness — padding → paddingDC₅DC₆: false by implication, tier V3, explicit witness — padding → padding → padding → 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.DC₆DCₙ, n > 6: false by implication, tier V3, explicit witnessDCJCnJacobian conjectureJC₁: proved (declared, tier V3)JC₁JC₂: open: nothing settles itJC₂openJC₃: refuted (declared, tier V0, explicit witness)JC₃JC₄: false by implication, tier V0, explicit witness — paddingJC₄JC₅: false by implication, tier V0, explicit witness — padding → paddingJC₅JC₆: false by implication, tier V0, explicit witness — padding → padding → paddingJC₆JCₙ, n > 6: false by implication, tier V0, explicit witnessJC
Open: PC₁, DC₂, JC₂. Refuted with an explicit witness: PCₙ for n ≥ 3; DCₙ for n ≥ 3; JCₙ for n ≥ 3. Refuted by implication, no witness exhibited: none. Claimed, not yet checked: PC₂ (a counterexample), DC₁ (a proof). Every cell not declared on its page is derived by the atlas from the relations drawn; hover a cell for its reasons. proved refuted, witnessed refuted by implication open claimed (V4) · a ring marks V0 · arrows point from the stronger statement to what it implies

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∈Q[x,q,p,z]R,T,D,S\in\mathbb Q[x,q,p,z] with R=x(2−3xq)R=x(2-3xq), in his convention {p,x}={z,q}=1\{p,x\}=\{z,q\}=1. The map (x,q,p,z)↦(R,T,D,S)(x,q,p,z)\mapsto(R,T,D,S) is a non-automorphic Poisson endomorphism whose fiber over (0,18,0,0)(0,\tfrac18,0,0) is exactly three rational points: (0,0,124,−18)(0,0,\tfrac1{24},-\tfrac18), (1,23,24796,−8964)(1,\tfrac23,\tfrac{247}{96},-\tfrac{89}{64}) and (−1,−23,24796,−8964)(-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 DD 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.