Poisson Conjecture

The Poisson conjecture asserts that every endomorphism of the canonical polynomial Poisson algebra over a field of characteristic 0 is an automorphism. Here the endomorphisms must preserve the canonical Poisson bracket. On , this Poisson bracket is determined by and zero Poisson brackets between the variables or between the variables. The integer counts canonical pairs, so the algebra has polynomial generators.

The conjecture is closely related to the Jacobian conjecture and the Dixmier conjecture. Preserving the canonical Poisson bracket forces the associated polynomial map to have constant Jacobian determinant 1, but global invertibility is the additional requirement.

Long (2026) gave an explicit noninvertible polynomial map preserving the Poisson bracket with two canonical pairs, hence four variables. One coordinate is , and a fiber of the map has three points. This refutes the conjecture for and, by adjoining unchanged canonical pairs, for ; the case remains open. The paper also gives a counterexample to the Dixmier conjecture for the fourth Weyl algebra. ChatGPT supplied the construction with additional AI auditing, and Long checked its proof.