Automorphism property of Poisson endomorphisms in rank two
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Must an endomorphism preserving the canonical Poisson bracket on the characteristic-zero polynomial algebra P₂ be an automorphism? Rank two means two canonical pairs, hence four variables.
Why it matters and what counts as progress
Why it matters: A rank-two counterexample would lower the number of canonical pairs needed for failure of the Poisson endomorphism conjecture.
What counts as progress: Explicit bracket-preserving polynomials with a proved noninvertibility obstruction, or an automorphism theorem for all rank-two endomorphisms.
How the problem developed
- 2007 · Equivalence [1]
Pascal Kossivi Adjamagbo and Arno van den Essen Related the Poisson, Jacobian and Dixmier conjectures, framing bracket-preserving endomorphisms within a wider polynomial-automorphism problem.
- 22 Jul 2026 · Reported claim [2]
Christopher D. Long Gave an explicit nonautomorphic Poisson endomorphism with two canonical pairs and stated that the construction extends to every rank at least two.
AI contribution
An Explicit Counterexample to the Rank-Two Poisson Conjecture
Long gives explicit rational-coefficient polynomials in four variables defining a Poisson endomorphism of the canonical algebra P₂ in characteristic zero that is not an automorphism. An exact three-point fiber proves noninjectivity; extending the map by identity gives examples for every Pₙ with n≥2.
Sources
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Publication dates and editorial changes.
ChatGPT-assisted map challenges the rank-two Poisson conjecture