Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverse
Assessment
Available evidence weighs against the claim.
This is the Jacobian conjecture, posed by Keller in 1939 and open for 87 years: that a polynomial self-map of complex n-space whose Jacobian determinant is a nonzero constant must have a polynomial inverse. In July 2026 the conjecture was refuted by an explicit counterexample announced by Levent Alpöge, developed with the AI model Claude Fable 5: a polynomial map of ℂ³ with constant Jacobian determinant −2 that is generically three-to-one, and therefore not invertible at all. The example is short enough to check by hand or by computer algebra, and its correctness has been confirmed independently across the mathematical community, including a detailed analysis by Terence Tao and a follow-up preprint constructing further examples. Since a three-dimensional counterexample extends to any higher dimension by adjoining identity coordinates, the conjecture fails in every dimension three and above.
The refutation does not reach the plane. The original two-dimensional case, the form in which Keller first posed the problem, remains open, and no known reduction brings the three-dimensional example down to two variables. The universal claim as stated is false; what survives of the question now lives in dimension two.
Full reasoning: the evidence and decisions behind this verdict
The verdict rests on the verified counterexample. The subclaim that Alpöge's map on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one stands verified at high confidence on the strength of direct symbolic verification: the determinant identity and the three-point generic fibers were checked computationally, not taken on authority. A universal claim over all dimensions is falsified by one counterexample, and the extension to every dimension above three is the elementary device of adjoining identity coordinates, so the inference from the counterexample to the parent's falsity has no gap. The companion subclaim that the conjecture is false in every dimension n ≥ 3 is not yet assessed, but the parent's verdict does not depend on it; the single verified example suffices.
The source record is one-sided. Denying instances: the Secret Blogging Seminar announcement (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), Terence Tao's digestion post (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), The Conversation's explainer, the follow-up arXiv preprint (arxiv.org/abs/2608.00222), and Wolfram MathWorld's updated entry, which states the conjecture is false in dimension three and in every higher dimension while the plane case remains open. Two web searches during this pass, including one looking specifically for error reports, disputes, or retractions, found none: the only live discussions in the neighborhood concern attribution (how much of the discovery was the AI's) and the still-open plane case, neither of which bears on the parent's truth. The one instance recorded with an affirming stance, Tao's remark that the conjecture remains open in two dimensions and easy in one, in fact asserts only the status of the low-dimensional restrictions, not the universal claim, so it does not constitute credible support and does not push toward contested.
The remaining credence covers only the residual possibility that the verification consensus collapses, which the elementary, independently reproduced symbolic checks make very unlikely. What would change the conclusion: a demonstrated error in the determinant identity or the fiber count of Alpöge's map, together with the failure of the follow-up preprint's further examples. A resolution of the two-dimensional case in either direction would not change this verdict, since the universal claim is already false in dimension three.
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.
- contradictsthis argues against the parentsteward instructions →Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one ↗︎
- specifiesa more specific version of the parentsteward instructions →Every polynomial map of ℂ² with nonzero constant Jacobian determinant has a polynomial inverse ↗︎
- contradictsthis argues against the parentsteward instructions →The Jacobian conjecture is false in every dimension n ≥ 3 ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The conjecture remains open in two dimensions, and is easy to establish in one dimension.
Stated immediately after Theorem 2, the three-dimensional counterexample; the comments repeatedly stress that the planar case is unresolved.
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.
It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well).
Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture.
As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture. Specifically, let a=(1+xy)^3z+y^2(1+xy)(4+3xy), ...
It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open.
Explainer on the July 2026 refutation; asserts in its own voice that Alpöge's function falsifies the conjecture in every dimension above two.
It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite ...
Abstract of a follow-up preprint constructing families of counterexamples in dimensions greater than two; asserts in its own voice that the conjecture was refuted in dimension three by Alpöge's explicit map.
Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension, while the plane case remains open (Zhang 2026).
Reference-work entry on the Jacobian conjecture, updated after the July 2026 announcement; states in its own voice that the conjecture is false in dimension three and above while the plane case remains open.
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.