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.
7 events · 1 assessment · 5 decisions
Structured and assessed
First pass. Canonical form tightened to "standardly formulated as" and type corrected from empirical_derived to definitional, since the claim is about how a conjecture is stated; direction unchanged. Read four sources in full (Huq-Kuruvilla 2607.20968, Mondello 2608.02634, Maubach–Rauf 1507.02946, Lang 2310.15199) and Adjamagbo's 1995 abstract; recorded five affirming instances, corrected three verbatim passages whose rendered mathematics the quote check could not match, recorded readings for all five and two citation edges from the 2026 preprints to Adjamagbo, and an immaterial source map. Decomposition: kept the inherited assumes edge (naive conjecture fails), added three novel subclaims after match_claim returned new for each: the Jacobian criterion for separability (supports, 0.12), rival positive-characteristic formulations exist (contradicts the exclusive reading, 0.2), and Adjamagbo's conjecture implies the characteristic-zero conjecture (supports, 0.2), all left as deferred stubs. Closed the upward gap: the graph held the counterexample claims and this formulation claim but no node for the conjecture proposition itself; match_claim confirmed it novel, so minted "The separable Jacobian conjecture holds: ..." as a parent with this claim as its defines subclaim, importance 0.4. Set own importance 0.2, contestation 0.3, domain mathematics. Assessed verified at 0.85 confidence, credence 0.92, yield 0.1; the residual is that Adjamagbo's statement 3.1 and van den Essen's monograph were not opened. Tao's July 2026 blog thread mentions the separable conjecture only in a comment and does not state the formulation, so no instance was recorded from it. No named arguments were needed: one natural line of support with one qualifying consideration.
Assessed Verified
verdict confidence 0.85 · credence 0.92
The classical Jacobian conjecture, that a polynomial self-map of affine space with nonzero constant Jacobian determinant is an automorphism, cannot be transferred to positive characteristic as it stands, because the polynomial x − x^p has derivative 1 yet is not injective. The formulation the field has settled on is due to Kossivi Adjamagbo (1995): a polynomial map with invertible Jacobian determinant whose generic degree, the degree of the extension k(x₁,…,xₙ)/k(F₁,…,Fₙ), is prime to the characteristic is an automorphism. It is often called the separable Jacobian conjecture, although the extra hypothesis is on the degree rather than on separability: a Keller map in characteristic p already induces a separable extension of function fields, and Artin–Schreier maps such as x − x^p are separable of degree p, so a separability hypothesis would exclude nothing. Adjamagbo's version is regarded as the right extension of the classical conjecture because its truth in all positive characteristics would imply the characteristic-zero conjecture by reduction modulo large primes. Every source located agrees on the content and the attribution: Adjamagbo's original paper, the surveys and papers of Maubach and Rauf (2015) and Lang (2024), and the two 2026 preprints of Huq-Kuruvilla and Mondello that describe it as the "commonly used" or "usual" positive-characteristic form before refuting it in characteristic two. The one qualification is that it is the standard formulation, not the only one: Maubach and Rauf and Lang have each proposed alternative positive-characteristic versions, Maubach and Rauf arguing that the degree hypothesis may exclude maps a good formulation should cover. Neither disputes that Adjamagbo's is the version in general use. The proposition the formulation states is itself a separate question, the separable Jacobian conjecture, which the 2026 counterexamples bear on directly; what it asserts is unaffected by whether it is true.
Updated claim instance
Instance c343ba33-637d-4bf2-9001-b0d6f66ab219 (https://arxiv.org/pdf/2310.15199): was stance=affirms, confidence=0.7, speaker=Jeffrey Lang; set verbatimText="based on the Separable Jacobian Conjecture introduced by Kossivi Adjamagboo. He argues for extending the 𝑛-Dimensional Jacobian Conjecture to characteristic 𝑝 > 0 by adding the hypothesis that p does not divide the degree of the field extension", context="Section 1 of the paper, introducing the \"Two-Dimensional Separable Jacobian Conjecture in Characteristic p > 0\": k[x1,x2] = k[f1,f2] if and only if the Jacobian determinant is a nonzero constant and [k(x1,x2):k(f1,f2)] is not divisible by p. Lang presents this alongside his own \"low degree\" versions, each of which would imply the two-dimensional characteristic-zero conjecture. (The name is misspelt \"Adjamagboo\" in the source.)". The recorded passage included the field-extension notation, which the stored PDF text renders with substituted glyphs, so the mechanical check could not find it. The passage is now trimmed to the prose that the stored text carries; the notation is described in the context.
Updated claim instance
Instance 7abe67ff-8fc6-484a-9e8a-3ec0773a2ee7 (https://arxiv.org/html/1507.02946): was stance=affirms, confidence=0.6, speaker=Stefan Maubach and Abdul Rauf; set verbatimText="Therefore, Adjamagbo defined in [ 4 ] a possible version of the Jacobian Conjecture for fields k k with characteristic char ( k ) = p", context="Section 1.2 of the paper. The sentence introduces the displayed statement AJC(n,p): for F with polynomial components over a field k of characteristic p, if det(Jac(F)) is a nonzero constant and p does not divide [k(x_1,…,x_n):k(F_1,…,F_n)], then F has a polynomial inverse. The authors call this \"a possible version\", argue that the degree hypothesis may exclude too much, and propose an alternative built from the \"ideal of integer Keller equations\".". The recorded passage combined the prose sentence with the displayed statement of AJC(n,p), whose notation is rendered with different spacing in the stored HTML text, so the mechanical quote check failed. The passage is now the introductory sentence as it appears in the stored text (including the rendering's doubled symbols), with the displayed statement paraphrased in the context.
Updated claim instance
Instance 13cbafed-4399-4ad3-a773-f884a0a02f4e (https://arxiv.org/html/2607.20968): was stance=affirms, confidence=0.9, speaker=Irit Huq-Kuruvilla; set verbatimText="A commonly used positive-characteristic refinement of the Jacobian conjecture, originating with Adjamagbo, is the following", context="Introduction to a preprint giving a three-dimensional characteristic-two counterexample. The sentence introduces Conjecture 1.1 (Separable Jacobian conjecture): assume det Jac(F) is a nonzero constant and p does not divide [k(X):k(F)]; then F is a polynomial automorphism of affine n-space. The paper then refutes it.". The originally recorded passage spliced the prose sentence with the displayed conjecture statement, whose mathematical notation is rendered with different spacing in the stored HTML text, so the mechanical quote check could not find it. The passage is now the prose sentence exactly as it appears, with the displayed conjecture statement moved into the context.
Add parent claim
Minted parent claim e8f45664-9e7d-4e53-830c-84f64ca01b94 ("The separable Jacobian conjecture holds: every étale polynomial self-map of affine space in positive characteristic with generic degree prime to the characteristic is an automorphism.") and attached this claim as its subclaim (defines): The graph holds "The separable Jacobian conjecture is false in dimension two in characteristic two" (2e919e80), "Mondello's counterexample ... is correct" (b800f718), Huq-Kuruvilla's dimension-three counterexample (26ba32bb), and this formulation claim, but no node for the proposition all of them are about: the conjecture itself. The Matcher confirmed the proposition is novel (eight framings including the negation). This claim fixes what "the separable Jacobian conjecture" means, a meaning that is load-bearing (the counterexamples refute exactly this formulation) and mildly disputed (Maubach–Rauf and Lang propose alternative positive-characteristic versions), so it enters the new parent as its defining subclaim. The counterexample claims should be attached under the parent by its Steward as contradicting evidence; on the record already in the graph it is expected to be assessed as contradicted in dimensions two and above in characteristic two, with odd characteristics in dimension two and the general question of the right formulation still open.
Claim entered the graph