“The Jacobian conjecture is false.”

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).

This problem is now known as Keller's problem but is more often called the Jacobian conjecture. This conjecture is still open (1999) for all $n \geq 2$.

Alpöge's map disproves the Jacobian Conjecture — the assertion, open since Keller (1939) [2], that every polynomial map ℂⁿ → ℂⁿ with nonzero constant Jacobian determinant is a polynomial automorphism — in dimension 3, and hence, by a standard stabilization argument, in every dimension n ≥ 3 [3].

The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\mathbb{C}^{n}$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. … It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23)

The conjecture, open since 1939 in every dimension $n\geq 2$ (see [2] for background and reductions and [8] for a survey), was refuted in July 2026: Alpöge publicly announced [1] an explicit polynomial map $F\colon\mathbb{C}^{3}\to\mathbb{C}^{3}$ with $\operatorname{Jac}F=-2$ possessing a fiber of three points (the announcement credits the AI system Fable with the construction).

The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\mathbb{C}^{n}$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23)

As of the time this text was written, O.-H. Keller’s Jacobian Conjecture remains an unresolved and seemingly insurmountable problem.

In July 2026, L. Alpöge announced a counterexample to the Jacobian Conjecture in dimension three, credited to the AI system Fable [1].

The conjecture has been stated by Keller in 1939 and is still open.

Levent Alpöge at Harvard University wrote on X on 19 July that the Jacobian conjecture – which academics have spent decades trying to prove was true – is actually false, giving a tiny, 216-character counterexample as proof.

Despite this, no proof for mappings of degree 3 has been found, so the conjecture remains unproven.

Many new exciting results have been obtained in the past two decades, including the solution of Nagata's Conjecture, the complete solution of Hilbert's fourteenth problem, the equivalence of the Jacobian Conjecture and the Dixmier Conjecture, the symmetric reduction of the Jacobian Conjecture, the theory of Mathieu-Zhao spaces and counterexamples to the Cancellation problem in positive characteristic.

It was recently shown(using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well): … > Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus’s proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later.

We verify an explicit polynomial map F : C 3 → C 3 with constant Jacobian deter minant − 2 and a fiber containing three points, thereby disproving the Jacobian conjecture in dimension three and, by stabilization, in every dimension at least three.

Hence \(\mathrm{JC}(3)\) (the Jacobian Conjecture in dimension 3) is false, and adjoining identity coordinates gives counterexamples in every dimension \(n\ge 3\).

Hence \(\mathrm{JC}(3)\) (the Jacobian Conjecture in dimension 3) is false, and adjoining identity coordinates gives counterexamples in every dimension \(n\ge 3\).

In this paper we give an update survey of the most important results concerning the Jacobian conjecture : several equivalent descriptions are given and various related conjectures are discussed.

This conjecture, now 82 years old and still unsolved for n>1, can be viewed as a problem in differential geometry, algebraic geometry, commutative algebra, analysis, topology, and combinatorics.

In particular F is not injective, so it has no polynomial left inverse and hence no polynomial inverse.