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.
Alpöge's map sends (x, y, z) to (a, b, c), where a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), and c = 2x − 3x²y − x³z. Both parts of the claim are matters of finite computation and both check out. The determinant of the Jacobian matrix simplifies identically to −2, so the map satisfies the hypothesis of the Jacobian conjecture everywhere on ℂ³. Yet a generic point of the target has exactly three distinct preimages: solving a = α, b = β, c = γ for generic values reduces to a cubic with distinct roots in one variable, with the other two coordinates then determined uniquely. A generically three-to-one map is in particular not injective.
The conjunction is what makes the map significant: constant-Jacobian maps and non-injective maps are each easy to produce, but a polynomial map with both properties is a counterexample to the Jacobian conjecture, which had stood open since 1939. The computation is short enough that it has been reproduced independently many times since the July 2026 announcement, including in Terence Tao's working through of the example and in subsequent preprints; no source disputes either computation. Discussion in the community has moved to questions the claim itself does not settle, such as attribution of the discovery and the still-open two-dimensional case.
Full reasoning: the evidence and decisions behind this verdict
The verdict rests primarily on direct verification, performed symbolically rather than taken from any source. Computing the 3×3 Jacobian matrix of (a, b, c) with respect to (x, y, z) and expanding its determinant yields exactly −2, an identity in the polynomial ring, not a numerical approximation. For the fiber count, a lexicographic Gröbner basis of the system a = α, b = β, c = γ was computed at two independent generic rational targets, (7/3, −5/2, 11/7) and (1/5, 3/4, −2/9). In both cases the basis is in shape position: a degree-3 univariate polynomial in z with nonzero discriminant (three distinct roots), together with polynomials linear in x and in y, so each fiber has exactly three distinct points; all three preimages of the first target were substituted back and reproduce the target to within 10⁻¹² numerically. Since fiber cardinality of a dominant polynomial map is constant on a Zariski-open set, two generic degree-3 fibers, consistent with the entire published discourse, establish "generically three-to-one." Non-injectivity follows immediately.
The source record points the same way. The original Secret Blogging Seminar post (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) states both facts and notes the Jacobian "is easily checked to be −2." The Conversation's explainer (theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883) asserts the constant determinant of −2 and the merging of input points. Terence Tao's post (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) builds a Bass–Connell–Wright reduction on the example, presupposing and thereby endorsing its correctness, and an arXiv preprint (arxiv.org/html/2608.00222) states the conjecture was refuted in dimension three by Alpöge's map. Searches turned up no source denying either computation; the only live disputes in the neighborhood concern attribution (how much of the discovery was the AI system's) and scope (the two-dimensional case remains open), neither of which bears on this claim's truth.
What would change the conclusion: an error found in the polynomial identity det J = −2 (reproducible by anyone in minutes, so vanishingly unlikely at this point) or a demonstration that the announced polynomials differ from the ones circulating in the sources verified here. The residual uncertainty in the credence covers only transcription risk between the announcement and the circulating formulas, which independent reproductions have largely eliminated.
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.
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.
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.
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.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.