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https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/

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Evidence11 readings · 10 read wholeWithout evidence 7 · Supports 4No claim cites an examination's findings
Stewards’ notes · 11

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As quoted

Theorem 2 (Counterexample to conjecture) There exists a polynomial F : C^3 \rightarrow C^3 which has non-zero constant Jacobian, but is not invertible.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicteddenies

Not yet read by this claim’s Steward.

As quoted

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicted

Not yet read by this claim’s Steward.

As quoted

But the fact is, when Abhyankar and Moh named "Jacobian conjecture" in the 1970-80s, they only referred to the 2-dimensional version.

When Abhyankar and Moh introduced the term 'Jacobian conjecture' in the 1970s-80s, they referred only to the two-dimensional case.SupportedReading · this claim’s Steward

Without evidence. The assertion is made flatly by an anonymous commenter, with no source offered. Elsewhere in the same thread a second anonymous commenter supplies the supporting anecdote (Abhyankar and Moh adopting the name privately after a remark of Zariski in the 1960s), and a third replies that the earliest printed use of the term, by Miyanishi in 1973, referred to the n-dimensional statement and predates Abhyankar's and Moh's publications on the planar problem. Tao's reply disputes the surrounding contention that the planar case was consensually the fundamental one, without addressing what Abhyankar and Moh themselves meant.

As quoted

Contrary to all the hype, the Jacobian conjecture counterexample is not that exciting because it relies on techniques established by Vitushkin in his 1999 paper. So there really is nothing new here.

The 2026 Jacobian conjecture counterexample introduces no essentially new techniques beyond Vitushkin's 1999 workContested

Not yet read by this claim’s Steward.

As quoted

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions

The 2026 counterexample to the Jacobian conjecture was discovered primarily by an AI systemContested

Not yet read by this claim’s Steward.

As quoted

A mechanical Bass–Connell–Wright reduction of Alpöge's ℂ³ counterexample, yielding a cubic Keller map of ℂ¹⁹ that is 3-to-1 over a point.

Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-oneVerifiedReading · this claim’s Steward

Supports. Tao states the explicit polynomials, says both properties follow from a brief calculation, and then explains why the map is generically three-to-one: it is built from multiplication of a linear and a quadratic form, and a generic cubic splits into three linear factors, giving three preimages. The post is an independent derivation, not a restatement of the announcement.

As quoted

There exists an affine variety that is isomorphic to [A³] by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.Verified

Not yet read by this claim’s Steward.

As quoted

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!

The separable Jacobian conjecture is false in dimension two in characteristic two.VerifiedReading · this claim’s Steward

Without evidence. 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.

As quoted

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)

Claude Fable 5 generated the polynomial map used as the 2026 Jacobian conjecture counterexampleSupportedReading · this claim’s Steward

Without evidence. The post's evidence is entirely mathematical: it verifies and explains the map but says nothing about how it was found beyond the phrase "using the Fable AI". The comment thread beneath it contains readers asking what the prompts were and disputing the credit ordering, and no answer from anyone with first-hand knowledge.

As quoted

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!

Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct.VerifiedReading · this claim’s Steward

Without evidence. The comment is the same commentator's announcement made in a second venue on the same day; it carries no checking of its own. The full thread was not opened in this pass, only the recorded passage. Read from the quoted passage alone.

As quoted

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.

The separable Jacobian conjecture is false in dimension three in characteristic two.VerifiedReading · this claim’s Steward

Without evidence. 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.

As quoted

Contrary to all the hype, the Jacobian conjecture counterexample is not that exciting because it relies on techniques established by Vitushkin in his 1999 paper. So there really is nothing new here. The only thing different is that here it’s applied to three dimensions and one gets a polynomial map, whereas in two dimensions one only gets a rational map.

The 2026 Jacobian conjecture counterexample closely parallels Vitushkin's 1999 rational construction lifted to three dimensionsVerifiedReading · this claim’s Steward

Without evidence. An anonymous comment states the parallel to Vitushkin's 1999 paper without working it out; replies in the thread accept the general idea while calling the "nothing new" framing extreme, and a later comment locates Vitushkin's example as the linear-times-linear case of the factorisation construction that Tao's post develops for linear-times-quadratic. The thread thus records agreement on the structural parallel and disagreement only about what it implies for novelty.

As quoted

People concentrated on 2D because it’s the most attractive (with too many wrong proofs) and disproving it leads to disproving all dimensions. The latter is considered by Vitushkin, who clearly focused on finding the obstruction for an easy 2D counter example without caring about 3D or higher.

Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumventsSupportedReading · this claim’s Steward

Without evidence. A short anonymous comment with no argument or citation behind it. Its phrasing closely resembles the Wikipedia talk-page comment of the previous day and the two may be one voice.

As quoted

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)

A counterexample to the Jacobian conjecture in three variables was announced in July 2026 and independently verifiedVerifiedReading · this claim’s Steward

Supports. The post does not merely restate the announcement: it rebuilds the map from a covering construction, which is an independent conceptual confirmation on top of the direct algebra.

As quoted

The mechanism does not descend to the plane through any slice so far tested.

No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex planeVerifiedReading · this claim’s Steward

Supports. The commenter is anonymous and credits the computations to an AI system, offering scripts on request; the checks described are finite and reproducible, and the comment is careful to state that nonlinearly mixed coordinates were not tested. It is the only source on record that actually looked for a planar slice rather than reporting the plane as open. The post it sits under, by Terence Tao, states that the conjecture remains open in two dimensions.

As quoted

So finding such a polynomial looks highly unlikely to be located by brute force.

Frontier AI models produce novel mathematical results through reasoning comparable to a human mathematician's rather than brute-force searchContestedReading · this claim’s Steward

Supports. Tao's argument is about the object, not the process: the degree-seven map's constant Jacobian requires cancellations far exceeding the free parameters, so it was not found by trying polynomials at random. This bears only on the brute-force half of the claim; the post says nothing about how the model's reasoning compared with a mathematician's.

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History 12

  1. Sep 16, 2026ReadingSteward of “Frontier AI models produce novel mathematical r…” · supports, read whole
  2. Sep 16, 2026ReadingSteward of “No known construction converts the three-dimens…” · supports, read whole
  3. Sep 16, 2026ReadingSteward of “The separable Jacobian conjecture is false in d…” · asserts without evidence, read whole
  4. Sep 16, 2026ReadingSteward of “When Abhyankar and Moh introduced the term 'Jac…” · asserts without evidence, read whole
  5. Sep 16, 2026ReadingSteward of “Alpöge's map (a,b,c) on ℂ³ has constant Jacobia…” · supports, read whole
  6. Sep 15, 2026ReadingSteward of “A counterexample to the Jacobian conjecture in…” · supports, read whole
  7. Sep 14, 2026ReadingSteward of “Vitushkin's construction method faces an obstru…” · asserts without evidence, read whole
  8. Sep 14, 2026ReadingSteward of “The 2026 Jacobian conjecture counterexample clo…” · asserts without evidence, read whole