The separable Jacobian conjecture is false in dimension two 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 form, 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 Romy Mondello posted a ten-page preprint exhibiting an explicit counterexample to this statement in dimension two over the algebraic closure of the field with two elements: the map F = (P, Q) with P = x + x²y + x⁴ + x⁶y² and Q = y + x⁵ + x⁶y + x⁷y² + x⁸y³. Its Jacobian determinant is identically 1, the three distinct points (0,1), (1,0) and (1,1) all map to (0,1), and the induced extension of rational function fields has degree exactly 3, which is odd and therefore both prime to the characteristic and separable.
Every load-bearing step is a finite exact computation, and the correctness of Mondello's construction has been confirmed by independent computer algebra: the determinant, the collision, and the elimination cubic b²x³ + (a³ + ab + 1)x + a that fixes the generic degree at 3 all reproduce. The map was found by restricting Huq-Kuruvilla's dimension-three characteristic-two counterexample to a fibre, so the same mechanism is now known to work in dimensions two and three. The preprint has not been peer reviewed and carries a Lean 4 formalization that its author presents as a receipt rather than a substitute for review, but no source disputes the construction and nothing short of a computational error, which independent checks exclude, would overturn it.
The result says nothing about the classical conjecture over the complex numbers, where the two-dimensional case remains open; it also leaves open the two-dimensional case in odd characteristics. The one residual qualification is definitional: if the discourse were to settle on a characteristic-p formulation other than Adjamagbo's, the claim would need restating rather than being false.
Full reasoning: the evidence and decisions behind this verdict
The previous assessment stopped at supported because the preprint (arxiv.org/abs/2608.02634) could not be opened and the exact characteristic-p formulation refuted was not stated by the sources then read. Both gaps are now closed. The preprint's title and abstract were read directly; the title is "A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two", and the abstract states the theorem, the map, the three-point collision, [k(x,y):k(P,Q)] = 3, and the separability of the extension. It also records that the map was derived from a coordinate-permuted form of Huq-Kuruvilla's three-variable map and that a Lean 4 formalization and an independent replay are documented in an appendix with explicit scope limits.
Independent verification in this pass: the Jacobian determinant of (P, Q) computed symbolically and reduced modulo 2 is exactly 1; the images of (0,1), (1,0), (1,1) modulo 2 are all (0,1); a lexicographic Gröbner basis of the ideal (P − a, Q − b) over GF(2) with y eliminated yields the single cubic b²x³ + (a³ + ab + 1)x + a in x, together with a generator linear in y, so the generic fibre has exactly three points and the function-field degree is 3. Degree 3 in characteristic 2 forces separability, since any inseparable extension has degree divisible by 2. This agrees with the recomputations recorded on the correctness of Mondello's counterexample, now verified at confidence 0.92, which additionally checked the paper's hidden-cubic recovery and irreducibility argument. The supporting subclaim, the separable conjecture's failure in dimension three in characteristic two, is likewise verified with independent computation, and its relation to the present claim is one of lineage, since Mondello's map is a fibre of Huq-Kuruvilla's.
Instances: three, all affirming. The primary is the preprint itself (Mondello, 29 July 2026). The two others are comments by David Speyer, one on the Secret Blogging Seminar thread (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) and one on Terence Tao's thread (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/, 6 August 2026); both relay the preprint without independent checking and are one voice. Two web searches for the preprint, its identifier, and terms indicating error or retraction returned only the preprint's arXiv pages, the related characteristic-zero preprint (arxiv.org/abs/2608.00222), and the Tao thread; no dissent, correction, or retraction was found. The support therefore rests entirely on the preprint's computations, which is why the verdict is grounded in reproducing them rather than in counting repetitions.
Weighing: the requires dependency is verified and independently recomputed here; the supporting dependency is verified; the instance set is lopsided toward affirmation with no counterweight in the discourse and none manufactured. Verified, credence 0.97. What would change it: a demonstrated computational error (excluded by two independent reproductions), or a shift in the discourse to a different characteristic-p formulation, such as the Maubach-Rauf proposal, which would make the claim ill-stated rather than false. Peer-reviewed publication would add little beyond what the direct computation already establishes.
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 →Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The separable Jacobian conjecture is false in dimension three in characteristic two. ↗︎ · shared subclaim
Provenance
Where this claim has been said, linked to its canonical form.
All support traces to a single primary document, Romy Mondello's July 2026 preprint, which states and proves the result by finite computation over the field with two elements. The two blog comments recorded are both by David Speyer announcing that preprint and add no checking of their own, so they are one voice relaying the paper rather than independent confirmation. The claim's standing therefore rests on the preprint's computations, which have been reproduced independently, rather than on the number of places it has been repeated.
And here is a reward for Huq-Kuruvilla's work — the two dimensional case in characteristic two! That's really exciting. Maybe we'll see the two dimensional case in some other characteristics soon!
Comment thread on Terence Tao's post digesting the July 2026 Jacobian conjecture counterexample; Speyer reports Mondello's two-dimensional characteristic-two counterexample (arXiv:2608.02634) as resolving that case, crediting Huq-Kuruvilla's dimension-three characteristic-two preprint as its origin.
Asserted without evidence of the source's own. A second announcement by the same commenter, David Speyer, in Terence Tao's comment thread; it restates the news of the preprint and adds no verification of its own.
Romy Mondello just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2
David Speyer: 'Just in case someone is relying on this comment thread for their main source of news about the Jacobian conjecture, Romy Mondello just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2 ... I believe this is the first 2-dimensional result we've seen.'
Asserted without evidence of the source's own. David Speyer's comment on his own blog thread reports the posting of Mondello's preprint and calls it the first two-dimensional result; it relays the preprint rather than verifying it.
Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two.
Abstract of the preprint exhibiting the explicit map F=(P,Q) over the algebraic closure of F_2 with Jacobian determinant 1, a three-point collision at (0,1), and generic degree 3; the paper is the primary source for the result.
The source's own evidence bears what it asserts. The preprint is the origin of the result. Its central claims are finite exact computations over the field with two elements, and the Jacobian determinant, the three-point collision and the generic degree of three have each been reproduced independently. Worth reading closely: It is the only primary source; it states precisely which characteristic-p formulation (Adjamagbo's separable one) is refuted, which is what the claim's canonical wording depends on.
How these sources relate
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ restates https://arxiv.org/abs/2608.02634, stating it more strongly than that document supports. The comment relays the preprint's own headline result and links it. Fidelity is recorded as strengthened only in the mild sense that the comment drops the word "separable", stating the result for the Jacobian conjecture in characteristic 2 without naming the formulation; the preprint is careful to name Adjamagbo's separable form. In context (the naive statement being trivially false) readers understand the substantive form, so the drift is small.
- https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ restates https://arxiv.org/abs/2608.02634, faithfully. The comment announces the preprint's result and credits its lineage from Huq-Kuruvilla's three-dimensional example exactly as the preprint itself does; it adds no evidence 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.
Assessment history
1 status change over 2 assessments. full history →
Cite this claim: a formal citation with its evidence attached
Contribute
Every judgment on this page is open to challenge. A contribution is evaluated on its merits by the reviewer; if it succeeds the page changes, and if it does not, the reasons are stated. Either way the exchange becomes part of the claim’s public record.
The attention this claim received was paid for by a funded mandate. Funding buys only scheduling: it can make an assessment happen sooner, or reach deeper into a subtree. It has no influence on what the assessment concludes, and none on which claims enter the graph; assessments run under the same public standards whoever pays, funders never see or shape a verdict before anyone else, and mandates that attempt to steer conclusions are refused.
Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.