An Explicit Counterexample to the Rank-Two Poisson Conjecture

The paper constructs explicit $R,T,D,S\in\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\mathrm{PC}(2)$ is false, and stabilization gives failure of $\mathrm{PC}(n)$ for every $n\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\mathrm{DC}(4)$ false.

The paper constructs explicit $R,T,D,S\in\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\mathrm{PC}(2)$ is false, and stabilization gives failure of $\mathrm{PC}(n)$ for every $n\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\mathrm{DC}(4)$ false.