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https://arxiv.org/html/2608.02634

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In positive characteristic the Jacobian conjecture is standardly formulated as: étale polynomial self-maps of affine space with generic degree prime to the characteristic are automorphisms.
Without evidence. The preprint adopts the degree-prime-to-p formulation explicitly as the convention it works under, attributing it to Adjamagbo and citing Wang, Maubach and Rauf, and Lang for it. It offers no argument that this is the standard version, which is ordinary for a paper whose contribution is the counterexample. Sep 16, 2026

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Text253 passages · 1 marked

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Report GitHub Issue6 passages

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Abstract0 passages

1 Introduction1 passage

2 The polynomial certificate0 passages

3 Reconstruction and transcendence degree0 passages

4 Irreducibility over the actual target field0 passages

5 Degree, minimal polynomial, and separability0 passages

6 The étale graph presentation0 passages

7 The geometric generic fiber0 passages

8 Collision and failure of isomorphism0 passages

9 A direct elimination certificate0 passages

10 Structural derivation from the three-variable map0 passages

11 Proof of the main theorem0 passages

A Scope of the Lean formalization0 passages

A.1 Historical local kernel receipt0 passages

A.2 Independent Harmonic Aristotle replay0 passages

References0 passages

License: arXiv.org perpetual non-exclusive license2 passages

Romy Mondello0 passages

Abstract5 passages

1 Introduction10 passages

Over k = 𝔽 ¯ 2 k=\overline{\mathbb{F}}_{2} , define8 passages

P , Q P,Q are algebraically independent and11 passages · 1 marked

In the formulation used here, the separable Jacobian conjecture is the conjectural implication that an étale polynomial endomorphism of affine space whose finite generic degree is prime to the characteristic must be an automorphism. The theorem below exhibits the failure of that implication in dimension two.

In positive characteristic the Jacobian conjecture is standardly formulated as: étale polynomial self-maps of affine space with generic degree prime to the characteristic are automorphisms.VerifiedReading · this claim’s Steward

Without evidence. The preprint adopts the degree-prime-to-p formulation explicitly as the convention it works under, attributing it to Adjamagbo and citing Wang, Maubach and Rauf, and Lang for it. It offers no argument that this is the standard version, which is ordinary for a paper whose contribution is the counterexample.

2 The polynomial certificate0 passages

Termwise differentiation gives3 passages

Hence3 passages

The values at the three indicated 𝔽 2 \mathbb{F}_{2} -points are2 passages

Set1 passage

Expanding in characteristic two gives2 passages

The two recovery identities are12 passages

Define9 passages

3 Reconstruction and transcendence degree0 passages

Let26 passages

4 Irreducibility over the actual target field4 passages

Write8 passages

Coefficient comparison in Q Q gives4 passages

5 Degree, minimal polynomial, and separability12 passages

At the hidden root, ( 3 ) gives2 passages

6 The étale graph presentation8 passages

The relative Jacobian matrix of this presentation is4 passages

7 The geometric generic fiber4 passages

There is an explicit bijection14 passages

8 Collision and failure of isomorphism0 passages

Let2 passages

9 A direct elimination certificate6 passages

10 Structural derivation from the three-variable map4 passages

Then the map7 passages

This is a polynomial coordinate system. The inverse formulas are1 passage

On the target make the linear change1 passage

Because3 passages

The fiber c = 0 c=0 is therefore preserved. Substituting5 passages

11 Proof of the main theorem2 passages

Appendix A Scope of the Lean formalization0 passages

The accompanying Lean 4 development fixes21 passages

A.1 Historical local kernel receipt2 passages

A.2 Independent Harmonic Aristotle replay2 passages

Acknowledgments1 passage

References27 references

Instructions for reporting errors7 passages

Major funding support from0 passages

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On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristicbook chapter19951 claimCites it faithfully, on 1 claim.

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