Rank-two Poisson Conjecture

The conjecture asks whether every polynomial map preserving the canonical symplectic Poisson bracket must be invertible. A 2018 note relates the two-variable case to the Jacobian conjecture without settling it.

July 2026 claimed counterexample

On July 22, 2026, Christopher D. Long submitted An Explicit Counterexample to the Rank-Two Poisson Conjecture to arXiv. Its abstract claims explicit polynomials defining a Poisson endomorphism of the algebra with two canonical pairs that is not an automorphism, and says the construction extends to every $n \ge 2$; the associated map of $\mathbb{A}^4$ allegedly has Jacobian determinant one but a three-point fiber. This would settle the problem negatively, but the claim is unverified.

Current status (as of September 2026): A preprint claims a counterexample for the rank-two case and for all $n \ge 2$, but the conjecture remains unverified rather than settled.