Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The map in question is the explicit polynomial self-map of ℂ³ announced by Levent Alpöge in July 2026 as a counterexample to the Jacobian conjecture: a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), c = 2x − 3x²y − x³z. Both properties stated in the claim are finite computations that anyone with a computer algebra system can reproduce in seconds, and they have been reproduced directly for this assessment. Expanding the determinant of the 3×3 Jacobian matrix gives the constant −2 as a polynomial identity. Solving a = α, b = β, c = γ for generic rational targets yields exactly three distinct preimages each time, and explicit collisions such as (0, 0, −1/4) and (1, −3/2, 13/2) both mapping to (−1/4, 0, 0) confirm the failure of injectivity concretely.
The three-to-one behaviour also has a conceptual explanation, worked out in the discussion on the Secret Blogging Seminar and in Terence Tao's account: the map is, up to a change of coordinates, the multiplication of a linear form by a quadratic form, and a generic binary cubic factors into three linear factors in three different ways. Over special loci the fiber degenerates (to two points, or to fewer distinct points where roots coincide), which is why the claim is stated generically.
No source disputes either computation. The live discussions around the counterexample concern how it was found, how much credit belongs to the AI system involved, and whether the two-dimensional conjecture survives; none of those bears on the two facts asserted here.
Full reasoning: the evidence and decisions behind this verdict
This pass re-verified both facts from the polynomials as printed in the Secret Blogging Seminar post (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) rather than relying on the earlier pass. The symbolic determinant of the Jacobian matrix expands to exactly −2. A lexicographic Gröbner basis of a = α, b = β, c = γ at two fresh generic rational targets, (7/3, −5/2, 11/7) and (2/9, −13/4, 5/11), is in shape position with a degree-3 univariate polynomial of nonzero discriminant, so each fiber has three distinct points; all six preimages reproduce their targets to within 10⁻¹⁰. Since fiber cardinality of a dominant polynomial map is constant on a Zariski-open set, this establishes generic three-to-one-ness. The explicit collisions published by John D. Cook (www.johndcook.com/blog/2026/07/21/jacobian-conjecture/) and by a commenter there, (0,0,−1/4) and (1,−3/2,13/2) to (−1/4,0,0), and (0,1,−4) and (−2,1,2) to (0,1,0), were checked and hold; the full fiber over (−1/4,0,0) is {(−1,3/2,13/2), (0,0,−1/4), (1,−3/2,13/2)}.
The source record is uniformly affirming and includes independent derivations, not only restatements. Tao's post (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) states both properties and explains the three-fold fibers via the multiplication map from linear × quadratic forms to cubic forms. The Secret Blogging Seminar post derives cubic elimination relations in each variable, consistent with degree three. The Conversation's explainer and its ScienceDaily syndication restate the announcement without computation and count as one voice. MathWorld and Wikipedia entries updated in August 2026 record the counterexample and note that its correctness is easy to verify with a computer algebra system. A search for any error, correction, or retraction found none.
One recording note: the Secret Blogging Seminar instance's passage does not match the stored copy of the page mechanically because the stored text strips the inline formulas ("the Jacobian of is easily checked to be ."); the recorded passage is the sentence with the formulas restored and is not a misquote.
What would change the conclusion: only a demonstration that the circulating polynomials differ from what Alpöge announced, and since every source examined prints the same three polynomials and the computed properties are exactly those announced, that risk is negligible. The residual credence gap reflects nothing more than this transcription possibility.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
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.
Blog post working through the structure of the counterexample; treats Alpöge's ℂ³ map as a genuine counterexample (constant Jacobian, multivalent fibers) and derives a cubic-homogeneous reduction from it.
The source's own evidence bears what it asserts. 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. Worth reading closely: It supplies the conceptual reason for the three-to-one behaviour, which the bare computation does not.
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.
Follows the explicit definition of the polynomials a, b, c in variables x, y, z.
The source's own evidence bears what it asserts. 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. Worth reading closely: It is the primary written record of the polynomials as announced; anyone checking transcription of the map against later sources should start here. The quoted passage was not found in the stored copy of this source.
He found an example of a function in three dimensions which has a constant Jacobian determinant of -2, and which moves multiple input points to the same output point, so it is not reversible.
Explainer on Alpöge's July 2026 refutation of the Jacobian conjecture, describing what the announced map does and why it falsifies the conjecture in dimensions above two.
Asserted without evidence of the source's own. The explainer, written by a mathematician, states the constant determinant of −2 and the merging of input points but offers no computation of its own, pointing readers to the fact that others verified the short formula. It is a faithful restatement of the announcement rather than independent evidence.
On July 19, 2026, the conjecture was refuted in dimension three by an explicit counterexample announced by L. Alpöge.
arXiv preprint on counterexamples to the Jacobian conjecture in dimensions greater than two; asserts in its own voice that Alpöge's explicit map refutes the conjecture in dimension three.
Alpöge came up with a counterexample, a polynomial function from ℝ³ to ℝ³ with constant Jacobian determinant −2. The function is · It's a tedious but simple calculus exercise to show that the determinant equals −2 everywhere. The inverse function theorem says that a function is locally invertible at any point where the Jacobian determinant is non-zero, so Alpöge's function is locally invertible everywhere. However, the function takes on some values more than once. For example, (0, 0, −1/4) and (1, −3/2, 13/2) both map to (−1/4, 0, 0).
Blog post explaining why local invertibility does not imply global invertibility, using Alpöge's map as the illustration; asserts the constant determinant −2 and exhibits an explicit pair of colliding points (does not state the generic fiber size of three).
The source's own evidence bears what it asserts. The post states the constant determinant of −2 as a calculus exercise and backs non-injectivity with a checkable explicit collision, two points sent to (−1/4, 0, 0); a commenter adds a second collision. Both collisions reproduce when substituted into the announced polynomials. The post speaks of the map over the reals and of "some values more than once" rather than of a generic fiber of three, so it bears a slightly weaker statement than the claim's.
He found an example of a function in three dimensions which has a constant Jacobian determinant of -2, and which moves multiple input points to the same output point, so it is not reversible.
ScienceDaily release carrying The Conversation's explainer text on the refutation; the sentence is identical to The Conversation's, so this is the same voice republished rather than an independent assertion.
Only a search excerpt of the release was seen; it repeats The Conversation's sentence word for word, so it adds no voice of its own.
How these sources relate
- https://theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883 restates https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/, faithfully. The Conversation restates the announced properties of the map and relies on the community's verification of the announced polynomials, linking out for it, rather than computing anything itself. The Secret Blogging Seminar post is the earliest written record of those polynomials and the verification discussion; the explainer's statement matches it without exaggeration (it says "multiple input points", weaker than three-to-one, which is not a distortion).
- Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture republishes https://theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883. The ScienceDaily release reproduces The Conversation's explainer sentence for sentence (the passage about the constant determinant −2 and colliding input points is verbatim identical); ScienceDaily routinely syndicates The Conversation's articles, and the release is dated two weeks after the original.
- 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
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.