https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/
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Stewards’ notes · 6
- The separable Jacobian conjecture is false in dimension two in characteristic two.
Without evidence. 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. Sep 16, 2026
- In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
Overstates. The comment gives the correct affine model of the complement (the slice where the equations of the hyperplane pullback and of the ramification divisor both equal one) but the specific third coordinate it states, 12vb² − 6ubc + 4b² + 2a − 3ac, does not define an isomorphism: on the chart it is quadratic in the remaining fibre variable and the resulting map has a non-constant Jacobian. The conclusion is nonetheless established within the same thread by other means: a point count over finite fields giving exactly q³, the same commenter's later argument writing the complement as an affine-line bundle over an affine-line bundle over a line, and a corrected coordinate change in a subsequent comment (u, 2vb − uc, −2D²a − 4Db + c with D = 2vb − uc) which does have a polynomial inverse. Sep 16, 2026
- Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
Supports. The post gives the three polynomials explicitly and states the determinant as something easily checked rather than deriving it, but its own elimination computations (a cubic relation in each remaining variable after eliminating the other two) and the geometric construction discussed in the comments account for the three-to-one behaviour. Recomputing from the printed polynomials reproduces both the constant determinant and the three-point generic fibers. Sep 16, 2026
- A counterexample to the Jacobian conjecture in three variables was announced in July 2026 and independently verified
Supports. The post gives the polynomials in full, so its assertion that the Jacobian is constant and the map three-to-one can be checked by anyone with a computer algebra system, and the comment thread records several such independent checks. Sep 15, 2026
- The separable Jacobian conjecture is false in dimension three in characteristic two.
Without evidence. 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. Sep 14, 2026
- Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct.
Without evidence. Speyer reports the preprint as a counterexample and offers no checking of his own in the comment; the shorthand "the 2-dimensional Jacobian conjecture in characteristic 2" omits the qualifier that the conjecture refuted is the separable, degree-prime-to-p formulation. Sep 14, 2026
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Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture.
Not yet read by this claim’s Steward.
Then the Jacobian of (a,b,c) is easily checked to be -2. However, the map (a,b,c) is generically three to one, not bijective.
Supports. The post gives the three polynomials explicitly and states the determinant as something easily checked rather than deriving it, but its own elimination computations (a cubic relation in each remaining variable after eliminating the other two) and the geometric construction discussed in the comments account for the three-to-one behaviour. Recomputing from the printed polynomials reproduces both the constant determinant and the three-point generic fibers.
Alpöge’s counterexample also refutes the Dixmier conjecture for A_n, n \geq 3.
Not yet read by this claim’s Steward.
Then (v,u,a,b,c) => (u, 2vb – uc, 12 v b^2 – 6 ubc + 4 b^2 +2a – 3ac) gives an isomorphism from the complement of R and H to A³.
Overstates. The comment gives the correct affine model of the complement (the slice where the equations of the hyperplane pullback and of the ramification divisor both equal one) but the specific third coordinate it states, 12vb² − 6ubc + 4b² + 2a − 3ac, does not define an isomorphism: on the chart it is quadratic in the remaining fibre variable and the resulting map has a non-constant Jacobian. The conclusion is nonetheless established within the same thread by other means: a point count over finite fields giving exactly q³, the same commenter's later argument writing the complement as an affine-line bundle over an affine-line bundle over a line, and a corrected coordinate change in a subsequent comment (u, 2vb − uc, −2D²a − 4Db + c with D = 2vb − uc) which does have a polynomial inverse.
the answer is no for every d ≥ 4 — d = 3 is the only case where the complement can be A^d, in any characteristic.
Not yet read by this claim’s Steward.
which constructs maps C^3 –> C^3 with constant Jacobian of every degree >= 3.
Not yet read by this claim’s Steward.
we construct an explicit integer polynomial in five variables (total degree 14, constant Hessian determinant 128, non-injective gradient) that refutes the Hessian conjecture HC_5.
Not yet read by this claim’s Steward.
HC_4 ==> JC_2.
Not yet read by this claim’s Steward.
Romy Mondello just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2
Without evidence. 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.
Alpöge’s Jacobian counterexample and Long’s SU(2) construction yield a 5-variable counterexample to the Generalized Vanishing Conjecture.
Not yet read by this claim’s Steward.
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 https://arxiv.org/abs/2608.02634 . I believe this is the first 2-dimensional result we’ve seen.
Without evidence. Speyer reports the preprint as a counterexample and offers no checking of his own in the comment; the shorthand "the 2-dimensional Jacobian conjecture in characteristic 2" omits the qualifier that the conjecture refuted is the separable, degree-prime-to-p formulation.
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.
Without evidence. 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.
As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture
Supports. The post gives the polynomials in full, so its assertion that the Jacobian is constant and the map three-to-one can be checked by anyone with a computer algebra system, and the comment thread records several such independent checks.
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History 7
- Sep 16, 2026ReadingSteward of “The separable Jacobian conjecture is false in d…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “In the 2026 Jacobian-counterexample constructio…” · overstates, read whole
- Sep 16, 2026ReadingSteward of “Alpöge's map (a,b,c) on ℂ³ has constant Jacobia…” · supports, read whole
- Sep 15, 2026ReadingSteward of “A counterexample to the Jacobian conjecture in…” · supports, read whole
- Sep 14, 2026ReadingSteward of “The separable Jacobian conjecture is false in d…” · asserts without evidence, read whole
- Sep 14, 2026ReadingSteward of “Mondello's counterexample to the two-dimensiona…” · asserts without evidence, read whole
- Aug 11, 2026Entered the graphEntered the graph; copy stored