The Separable Jacobian Conjecture in Characteristic 2

Adjamagbo's positive-characteristic refinement of the Jacobian conjecture asks that a polynomial endomorphism with unit Jacobian determinant whose induced function-field extension has degree prime to the characteristic be an automorphism. It is false: an explicit $F: \mathbb{A}_k^3 \to \mathbb{A}_k^3$ over any field of characteristic $2$ has Jacobian determinant identically $1$ and function-field degree $3$ , yet is not injective.

Result: Disproved(see note)
Status: Resolved
AI contribution: AI co-developed
Method: Construction
Field: Affine algebraic geometry
Posed by: Kossivi Adjamagbo (specific case)
Year posed: 1995
Years open: 31y
Solved: 2026-07-23
Model: ChatGPT 5.6 Sol
Vendor: OpenAI
Collaborators: Irit Huq-Kuruvilla
Verification: Site-confirmed
Publication: Preprint
Significance: 25 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

The counterexample to the original Jacobian conjecture does not specialize to characteristic 2; this is a modification that does.

What the AI did

The AI usage section states that the text is the result of a discussion with ChatGPT 5.6 Sol, with conversation transcripts available on request, and that the proof itself was verified as correct by the named author.

Verification

The counterexample is a three-line explicit polynomial map; the non-injectivity is witnessed by three distinct source points and the Jacobian determinant is a direct computation, both checkable by hand. Single-author arXiv note, not yet peer-reviewed.

Sources

PaperarXiv:2607.20968 - An Explicit Characteristic-2 Counterexample to the Separable Jacobian Conjecture
Independent workDimension-two counterexample, Lean kernel-checked (Mondello)
Problem recordAdjamagbo 1995 - Conjecture statement

Related entries

RelatedJacobian Conjecture