A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two

Abstract

Let k be the algebraic closure of F_2. We study the polynomial endomorphism F=(P,Q) of the affine plane, where P=x+x^2 y+x^4+x^6 y^2 and Q=y+x^5+x^6 y+x^7 y^2+x^8 y^3. Its Jacobian determinant is 1, while the three distinct points (0,1), (1,0), and (1,1) have the same image. We prove that [k(x,y):k(P,Q)]=3 and that the extension is separable. Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two. The proof uses explicit recovery from a hidden cubic, irreducibility over the actual target field, and a bridge between function-field embeddings and the geometric generic fiber. We also give an explicit graph presentation proving that F is etale and derive the map from a coordinate-permuted form of a three-variable map of Irit Huq-Kuruvilla. An appendix records the precise scope and evidence boundaries of a Lean formalization and an independent Harmonic Aristotle replay.

Disclosure

“ns. Neither receipt is a claim of human peer review, and the external replay does not replace the Lean kernel as the authority for the checked theorem. Acknowledgments Lean and Mathlib were used to formalize and kernel-check the theorem. OpenAI ChatGPT and Codex assisted with proof organization, code navigation, and preparation of the exposition. Harmonic Aristotle was used as a computationally separate replay environment. All mathematical claims remain the responsibility of the named au”

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Proof ideas or individual proof-step assistance

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8

Verified

2026-08-23T19:13:53.563439Z

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