An Explicit Counterexample to the Rank-Two Poisson Conjecture

Abstract

Let \[ {\mathcal P}_2={\mathbb C}[x,q,p,z] \] carry the canonical Poisson bracket determined by $ \{p,x\}=\{z,q\}=1 $ and by the vanishing of the other brackets between distinct generators. Here and throughout, ``rank two'' means two canonical pairs in the standard indexing of the canonical Poisson algebras; thus there are four polynomial generators and the Poisson tensor has geometric rank four. We give explicit polynomials \[ R,T,D,S\in{\mathbb Q}[x,q,p,z] \] satisfying \[ \{D,R\}=1,\qquad \{S,T\}=1, \qquad \{R,S\}=\{R,T\}=\{D,S\}=\{D,T\}=0, \] while \[ R=x(2-3xq). \] Consequently, the assignment $ (x,q,p,z)\mapsto(R,T,D,S) $ defines a Poisson endomorphism of ${\mathcal P}_2$ that is not an automorphism. This disproves the Poisson Conjecture for two canonical pairs, and hence for every number of canonical pairs at least two. The associated polynomial map of ${\mathbb A}^4$ preserves the canonical symplectic form, has Jacobian determinant one, and has an explicit fiber consisting of exactly three points. The proof uses a polynomial source coordinate system in which the symplectic identity reduces to three displayed coefficient identities. A separate appendix uses the same four output polynomials and their Hamiltonian duals to construct an explicit nonautomorphic endomorphism of the fourth Weyl algebra.