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.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim is definitional: it says how the Jacobian conjecture is standardly formulated in positive characteristic. It was checked against five sources, four read in full.
Adjamagbo's 1995 paper (abstract only; link.springer.com/chapter/10.1007/978-94-015-8555-2_5) identifies the one-variable failure x − x^p as having geometric degree a multiple of p, calls this "the only accident" that breaks the classical conjecture in characteristic p, and proposes excluding it to obtain "the right and universal formulation" in any characteristic. Maubach and Rauf (arxiv.org/html/1507.02946, section 1.2) reproduce the statement AJC(n,p) exactly as the claim has it: det Jac(F) a nonzero constant and p not dividing [k(x):k(F)] imply a polynomial inverse. Lang (arxiv.org/pdf/2310.15199, section 1) attributes the same hypothesis to Adjamagbo and states the two-variable version. Huq-Kuruvilla (arxiv.org/html/2607.20968) states it as Conjecture 1.1, "a commonly used positive-characteristic refinement ... originating with Adjamagbo", and Mondello (arxiv.org/html/2608.02634) as "the Adjamagbo, or separable, formulation", citing Adjamagbo, Wang, Maubach and Rauf, and Lang. All five instances affirm; none denies. The two 2026 preprints add no independent argument for the formulation's standing and rest on Adjamagbo's authority, so the support is one origin restated by four later voices, which for a claim about attribution and usage is the right kind of evidence.
Of the subclaims, the failure of the naive conjecture for x − x^p is the background every source starts from and is elementary (the derivative is 1; the map is p-to-one on the algebraic closure). The Jacobian criterion for separability is textbook (Lang's Proposition 1.4 gives a two-line derivation proof) and explains why the refinement is on the degree: since étale already gives separability, "separable Jacobian conjecture" is a name, not a description of the added hypothesis. The implication to the characteristic-zero conjecture is proved in two variables by Lang (Proposition 1.6) and credited to Adjamagbo in general by Maubach and Rauf; it is what makes this formulation a genuine extension rather than an arbitrary weakening. The consideration against is the existence of rival formulations: Maubach and Rauf's integer-Keller-equations version and Lang's low-degree versions. These are real and published, so "the Jacobian conjecture asserts" cannot be read as exclusive; the canonical form says "standardly formulated" for that reason, and under that reading the rival proposals do not weigh against the claim, since their own authors describe Adjamagbo's version as the baseline.
What was not done: Adjamagbo's statement 3.1 and his general-dimension reduction argument sit behind a paywall and were not read, and van den Essen's 2000 monograph, which the seed note and general knowledge say presents the same formulation, was not opened. A reader finding that Adjamagbo's own 3.1 differs materially from the degree-prime-to-p statement, or that a survey of the field names a different formulation as standard, would be the evidence that changes this verdict. The 2026 counterexamples do not: a conjecture's content does not change when it is refuted.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- assumesbackground the parent's framing takes as givensteward instructions →The naive Jacobian conjecture fails in positive characteristic: the polynomial x − x^p has derivative 1 but is not injective. ↗︎
- supportsthis provides evidence for the parentsteward instructions →A polynomial map in positive characteristic with nonzero Jacobian determinant induces a separable algebraic extension of rational function fields. ↗︎
- contradictsthis argues against the parentsteward instructions →Positive-characteristic formulations of the Jacobian conjecture other than Adjamagbo's degree-prime-to-p version have been proposed. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Adjamagbo's separable Jacobian conjecture in positive characteristic implies the classical Jacobian conjecture in characteristic zero. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
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.
Introduction to a preprint exhibiting a two-dimensional characteristic-two counterexample; the author attributes the formulation to Adjamagbo and cites Wang 1980, Maubach–Rauf 2017 and Lang 2025 for it, while flagging "in the formulation used here".
Asserted without evidence of the source's own. 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.
A commonly used positive-characteristic refinement of the Jacobian conjecture, originating with Adjamagbo, is the following
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.
Asserted without evidence of the source's own. The preprint calls the degree-prime-to-p version the commonly used positive-characteristic refinement and attributes it to Adjamagbo, resting the description on references rather than on argument. Its own mathematics concerns the counterexample, not the formulation.
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
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 source's own evidence bears what it asserts. Lang states the formulation faithfully and attributes it to Adjamagbo, and his Proposition 1.6 supplies the reason it counts as a genuine extension of the classical conjecture: in two variables it implies the characteristic-zero statement. He sets it beside his own low-degree versions, so the paper treats it as a leading formulation rather than the only one.
Therefore, Adjamagbo defined in [ 4 ] a possible version of the Jacobian Conjecture for fields k k with characteristic char ( k ) = p
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 source's own evidence bears what it asserts. Maubach and Rauf give Adjamagbo's statement precisely and explain why it is a natural response to the one-variable counterexample, so their account of the formulation is well grounded. They present it as one possible version rather than the definitive one, and the paper's main purpose is to propose an alternative, so this source is the best place to see the case that the standard formulation is not the only reasonable one. Worth reading closely: It is the clearest published argument that the degree-prime-to-p hypothesis may be the wrong refinement, and it states the main rival formulation; a reader weighing whether Adjamagbo's version deserves to be called the positive-characteristic Jacobian conjecture should see it.
But we could remark that the geometric degree of F, i.e. the dimension of the field F_p(X) over F_p(F), is a multiple of p. From our point of view, this fact is the only accident which could made the traditional formulation of the Jacobian Conjecture fall down in characteristic p. Hence, we think that it is sufficientce to avoid this accident to obtain the right and universal formulation of the classical Jacobian conjecture for the automorphisms of the algebras of polynomials in any number of polynomials over any domain of any characteristic (see its precise statement in 3.1).
The originating paper: Adjamagbo argues that excluding maps whose geometric degree is a multiple of p yields the correct formulation of the Jacobian conjecture in every characteristic. Only the abstract was read; the precise statement (3.1) sits behind a paywall.
Only the publicly visible abstract of Adjamagbo's 1995 paper was read; the precise statement (his 3.1) and the reduction argument sit behind a paywall. The abstract makes the motivating case from the one-variable example and announces the formulation, but whether the body carries the argument the later literature credits it with could not be checked here. Worth reading closely: It is the primary source for the formulation and for the claim that the positive-characteristic conjecture for all p implies the characteristic-zero conjecture; a reader with access should confirm the exact statement of 3.1 and whether the general-dimension reduction is proved there.
How these sources relate
- https://arxiv.org/html/2607.20968 cites in support On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic, faithfully. The preprint attributes the formulation to Adjamagbo and cites his paper (among others) as its source; the statement it gives matches Adjamagbo's abstract (exclude maps whose geometric degree is a multiple of p) and the statement Maubach and Rauf reproduce from the original. Only Adjamagbo's abstract was opened, not the body containing statement 3.1. Judged from the citing document alone.
- https://arxiv.org/html/2608.02634 cites in support On Separable Algebras over a U.F.D. and the Jacobian Conjecture in Any Characteristic, faithfully. The preprint names Adjamagbo as the source of the formulation and cites his 1995 paper together with Wang, Maubach and Rauf, and Lang. Its statement of the conjecture agrees with the original's abstract and with the later restatements. Only Adjamagbo's abstract was opened. Judged from the citing document alone.
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Created by claim_steward · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.