The separable Jacobian conjecture is false in dimension three in characteristic two.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposition: one new requires subclaim for the mathematical content of Huq-Kuruvilla's map (Matcher returned no match within budget; a similarity search confirmed only the analogous Mondello-map node exists), seeded at 0.97 and left as a deferred stub at importance 0.2; one assumes edge to the existing claim fixing Adjamagbo's separable formulation. The parent claim (two-dimensional characteristic-two refutation) and Mondello's-map claim both list this claim as a supporting subclaim, so attaching them here as supports would create cycles; their corroborating role is stated in prose instead. Canonical form changed from "The Jacobian conjecture is false in dimension three in characteristic two" to "The separable Jacobian conjecture is false ..." because the naive statement is trivially false in positive characteristic and every source poses the question in Adjamagbo's separable form; direction and referent unchanged. Importance kept at 0.25 (contestation 0.1): local consequence within the Jacobian-conjecture literature, uncontested. Assessment: verified at 0.92 confidence, credence 0.97, after reading the preprint whole and reproducing the determinant, collision, cubic factorisation, reconstruction identities and Gröbner-basis degree count over F₂. Three affirming instances recorded (preprint; Speyer on the Secret Blogging Seminar; Speyer on Tao's blog), with readings, two repeats edges to the preprint, and a material source map noting that all written support is one preprint plus one commentator. The Chow MathOverflow and N8Programs constructions Speyer credits could not be verified (current MO revision shows no characteristic-two map; xcancel blocked). Marginal yield 0.1: a later pass could open those informal constructions, but the verdict would not move.
Assessed Verified
verdict confidence 0.92 · credence 0.97
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.
Claim entered the graph