The separable Jacobian conjecture is false in dimension three in characteristic two.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
In positive characteristic the Jacobian conjecture cannot be stated naively, since maps such as x ↦ x + x² have Jacobian determinant 1 yet are two-to-one; the substantive version, due to Adjamagbo, asserts that a polynomial self-map of affine space with invertible Jacobian determinant and generic degree prime to the characteristic is an automorphism. In July 2026 Irit Huq-Kuruvilla posted a six-page preprint exhibiting an explicit counterexample to this statement in dimension three and characteristic two: the map (x + x²y, y + xz + x²yz, z + x²z²). Its Jacobian determinant is identically 1, the three points (0,1,0), (1,1,0) and (1,1,1) all map to (0,1,0), and the induced extension of rational function fields has degree exactly 3, which is odd and hence prime to 2. That the map has Jacobian determinant 1 and generic degree 3 yet is not injective is a finite exact computation, reproduced independently by computer algebra over the field with two elements: the determinant, the collision, the factorisation of the cubic UT³ + T² + VT + W that governs the extension, and a Gröbner-basis count of the generic fibre all agree with the preprint.
The result closes the one case that the characteristic-zero refutation of July 2026 left open in positive characteristic. Alpöge's three-variable map has Jacobian determinant −2, so it reduces to a counterexample to the separable conjecture in every odd characteristic but degenerates modulo 2, which is why a fresh construction was needed. Informal characteristic-two constructions by Timothy Chow and by the pseudonymous N8Programs circulated first; Huq-Kuruvilla's was the first written up cleanly, and Romy Mondello's later two-dimensional characteristic-two counterexample was found by restricting it to a fibre, which in turn yields a second three-dimensional example on padding with an identity coordinate. No source disputes the construction, and nothing beyond a computational error, which two independent checks exclude, would overturn it. The only residual qualification is definitional: if the discourse's characteristic-p Jacobian conjecture were taken to be a formulation other than Adjamagbo's, such as the one proposed by Maubach and Rauf, the claim would need restating rather than being false.
Full reasoning: the evidence and decisions behind this verdict
Direct verification of the primary source. Huq-Kuruvilla's preprint (arxiv.org/abs/2607.20968, full text at arxiv.org/html/2607.20968) was read whole. Its Theorem 1.2 asserts that F = (P,Q,R) = (x + x²y, y + xz + x²yz, z + x²z²) over any field of characteristic 2 has det Jac(F) = 1, [k(x,y,z):k(P,Q,R)] = 3, and is not injective, refuting Conjecture 1.1 (Adjamagbo's separable Jacobian conjecture: det Jac ∈ k× and p ∤ [k(X):k(F)] imply automorphism). Each part was recomputed with SymPy. Over Z the Jacobian determinant is 1 + 2xy + 2x²z + 4x³yz + 2x⁴z² + 2x⁵yz², so it is exactly 1 modulo 2. F(0,1,0) = F(1,1,0) = F(1,1,1) = (0,1,0) over F₂; the fibre over (0,1,0) has three points of F₂³ and every other image point has one. For the degree, a lex Gröbner basis of (P − a, Q − b, R − c) in F₂[x,y,z,a,b,c] with x > y > z > a > b > c has leading monomials in x,y,z of x, y and z³ once a,b,c are treated as field elements, so the generic fibre algebra has basis {1, z, z²} and dimension 3; since it is finite over k(a,b,c) and a domain, it is the field k(x,y,z), giving degree 3. The preprint's own route also checks: with U = P, V = Q + PR, W = R, the cubic D(T) = UT³ + T² + VT + W equals (xT + b)(cT² + dT + z) for b = 1 + x²z, c = 1 + xy, d = Q; the identities xd + bc = 1 and Res_T = 1 hold modulo 2; t = b/x is a root with D′(t) = 1/x; and the reconstruction x = 1/D′(t), z = a(t + a), y = Q + Pz holds identically, so k(x,y,z) = K(t). D is monic of degree one in W, hence irreducible in k[U,V,W,T], and primitive in T (coefficient 1 on T²), so irreducible over K by Gauss's lemma; degree exactly 3, odd, hence separable. Corollary 5.1 (padding by identity coordinates) extends the counterexample to every dimension at least 3.
Subclaims. Huq-Kuruvilla's map has Jacobian determinant 1 and generic degree 3 yet is not injective is the mathematical content and is established by the computations above. The positive-characteristic Jacobian conjecture asserts that an étale map of generic degree prime to p is an automorphism fixes what is refuted; the preprint states this formulation and cites Adjamagbo (1995), Kimura and Okuda (2005) and Maubach and Rauf (2017). The naive statement is trivially false in every positive characteristic (Artin-Schreier), so this is the only reading under which the claim is substantive, and it is the reading every source uses; the existence of alternative formulations is the reason verdict confidence is 0.92 rather than higher.
Instances and discourse. Three instances, all affirming: the preprint (23 July 2026); David Speyer's comment of 6 August 2026 on the Secret Blogging Seminar (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), which calls the preprint a "dimension 3, characteristic 2, counterexample" and credits earlier informal constructions by Timothy Chow (mathoverflow.net/a/513393/297) and N8Programs (xcancel.com/N8Programs/status/2079061392649302185); and Speyer's comment the same day on Terence Tao's blog (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), replying to an anonymous commenter who wrote that Adjamagbo's separable conjecture "just got disproven for all dimensions n > 1". Speyer's two comments are one voice restating the preprint. Earlier in the Secret Blogging Seminar thread (20 and 21 July) Lior Silberman observed that Alpöge's map, with Jacobian determinant −2, refutes the separable conjecture in every odd characteristic and asked whether anyone was looking at characteristic 2, and Speyer noted that the geometric construction works in all characteristics; the preprint itself notes that Alpöge's example "does not specialize" to characteristic 2. The MathOverflow page as fetched shows Chow's answer in its current revision, which gives an integer-coefficient Jacobian-1 variant of Alpöge's map but no characteristic-two map (that variant reduces to the identity modulo 2), so the informal Chow and N8Programs constructions could not be checked here; the N8Programs post was behind a bot check. An automated review site (Pith) lists the paper as "the math is right and the counterexample is real", faulting only its historical remark that no other counterexample was known; that page could not be read whole and is not recorded as an instance. Two web searches found no dissent, erratum, or retraction.
Weight of the support. All written support traces to one preprint, but the verdict does not rest on the announcements: it rests on the preprint's proof, which is elementary and was reproduced independently in full. Mondello's two-dimensional map (arxiv.org/abs/2608.02634), already verified on its own claim, is a fibre restriction of this map after coordinate permutation, so it corroborates the construction's soundness without being independent of it; padded by an identity coordinate it gives a second dimension-three counterexample.
What would change the verdict: an error in the degree computation (excluded by two independent methods) or a demonstration that the discourse's characteristic-p Jacobian conjecture refers to a formulation other than Adjamagbo's, which would make this a wording question rather than a refutation. Peer-reviewed publication would add little; the mathematics is already checked.
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.
- requiresa load-bearing premise: the parent is false without itsteward instructions →Huq-Kuruvilla's map (x+x²y, y+xz+x²yz, z+x²z²) in characteristic two has Jacobian determinant 1 and generic degree 3 yet is not injective. ↗︎
- assumesbackground the parent's framing takes as givensteward instructions →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. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The support rests on one six-page preprint by Irit Huq-Kuruvilla, which carries a complete elementary proof that its explicit three-variable map is étale, generically three-to-one and not injective in characteristic two; every step of that proof reproduces under independent computer algebra. The two other appearances are the same commentator, David Speyer, announcing the result in two blog threads on the same day without checking of his own, though he also credits two earlier informal constructions, by Timothy Chow on MathOverflow and by N8Programs on X, that could not be opened in a form showing the maps. A reader should open the preprint first; the blog comments add only that an expert in the area treats the result as sound and knows its history.
The first observation about passing to characteristic 2, dimension 3, has been recorded elsewhere; see Timothy Chow https://mathoverflow.net/a/513393/297 and N8Programs https://xcancel.com/N8Programs/status/2079061392649302185 . That said, I am in favor of results being recorded in clearly written versions posted to repositories like the arXiv, so good for Irit Huq-Kuruvilla for putting in the work.
Reply to an anonymous commenter who had written that Adjamagbo's separable Jacobian conjecture "just got disproven for all dimensions n > 1", citing Huq-Kuruvilla's and Mondello's preprints; Speyer treats the dimension-three characteristic-two counterexample as established and credits the earlier informal observations by Chow and N8Programs.
Asserted without evidence of the source's own. The comment is the same commentator's announcement made in a second venue on the same day and carries no checking of its own; its value is that an expert in the area treats the dimension-three characteristic-two result as established and knows its history. The anonymous comment it replies to asserts the stronger statement that the separable conjecture fails in every dimension above one, citing both preprints.
Interestingly, Mondello found his result by looking at Huq-Kuruvilla’s dimension 3, characteristic 2, counterexample https://arxiv.org/abs/2607.20968 . This was not the first counterexample with dimension 3, characteristic 2, ( https://mathoverflow.net/a/513393/297 , https://xcancel.com/N8Programs/status/2079061392649302185 ), but I believe it was the first one written up in a clean preprint on the arXiv.
Comment by the post's author in the thread on Alpöge's characteristic-zero counterexample, reporting Mondello's two-dimensional result and describing Huq-Kuruvilla's preprint as a dimension-three, characteristic-two counterexample, with earlier informal constructions by Timothy Chow (MathOverflow) and N8Programs (X).
Asserted without evidence of the source's own. Speyer describes the preprint as a dimension-three, characteristic-two counterexample without offering any checking of his own, and points to two earlier informal constructions of the same kind by Timothy Chow on MathOverflow and N8Programs on X. Earlier in the same thread Lior Silberman had noted that Alpöge's characteristic-zero map works in every odd characteristic and asked whether anyone was looking at characteristic two; the comment answers that question.
Since $2\neq 3$, this gives a counterexample to the usual Adjamagbo, or separable, formulation of the Jacobian conjecture in characteristic $2$.
The preprint's abstract, stating its main theorem: an explicit polynomial endomorphism of affine three-space over any field of characteristic two with Jacobian determinant 1 and function-field extension of degree 3 that is not injective, refuting the separable (Adjamagbo) Jacobian conjecture in characteristic 2 and dimension 3.
The source's own evidence bears what it asserts. The six-page paper carries a complete elementary proof. The Jacobian matrix is written out and its determinant is 1 because squares have zero derivative in characteristic two; three explicit points over the prime field collide; and a cubic UT³+T²+VT+W in the target coordinates factors over the source field as a linear times a quadratic with resultant 1, which yields a root generating the whole source field and, by Gauss's lemma, degree exactly three. Every identity reproduces under independent computer algebra over F₂. The author states the text arose from a conversation with a language model and that the proof was verified by the named author; the remark that no other counterexample was known is contradicted by earlier informal constructions that Speyer credits to Chow and N8Programs, but that historical point does not touch the mathematics. Worth reading closely: It is the primary source and the only one carrying a proof; it is short and fully checkable by hand or computer algebra.
How these sources relate
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ restates https://arxiv.org/abs/2607.20968, faithfully. The comment links the preprint directly and restates its result as a dimension-three characteristic-two counterexample, adding no evidence of its own. The shorthand omits "separable", but in the thread's context (Silberman's earlier question about the separable conjecture in characteristic two) the referent is the same formulation the preprint refutes.
- https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ restates https://arxiv.org/abs/2607.20968, faithfully. The comment replies to a post citing the preprint and treats its dimension-three characteristic-two counterexample as the written-up version of an observation earlier recorded informally by Chow and N8Programs; it adds no checking of its own.
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ and https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ share an author. Both recorded passages are comments by David Speyer (one on his own Secret Blogging Seminar thread, one on Terence Tao's thread) announcing the same preprint; they are one voice, not two.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.