Jacobian Conjecture from Wolfram MathWorld

Summary

The Jacobian conjecture asserts that a polynomial map having a nonzero constant Jacobian determinant is an automorphism. In the plane, first stated by Keller (1939), it says that a ring map of (the polynomial ring in two variables over the complex numbers) to itself that fixes and sends , to , , respectively, is an automorphism iff the Jacobian is a nonzero element of . The condition is easily shown to be necessary. The plane case remains open. There have been at least five published incorrect proofs and many incorrect attempts over the years. In November 2004, Hochster (2004) sent an email announcing a new proof by Carolyn Dean. However, this proof contained an error as well. In July 2026, Alpöge (2026) announced the following polynomial counterexample, which he credited to the AI system Fable. Writing , its coordinate polynomials are Its Jacobian determinant is the nonzero constant (4) but the three distinct points (5) have the same image, since (6) Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension (Zhang 2026). The Jacobian conjecture is one of Smale's problems. See alsoInvertible Polynomial Map, Jacobian, Polynomial Map, Smale's Problems Explore with Wolfram|Alpha ReferencesAbhyankar, S. S. Lectures on Expansion Techniques in Algebraic Geometry. Bombay, India: Tata Institute of Fundamental Research, 1977.Alpöge, L. X post, July 20, 2026. https://x.com/__alpoge__/status/2079028340955197566.Bass, H. "Conjecture jacobienne et opérateurs différentiels." Mém. Soc. Math. France, No. 38, 39-50, 1989.Bass, H.; Connell, E. H.; and Wright, D. "The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse." Bull. Amer. Math. Soc. 7, 287-330, 1982.Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, p. 330, 1993.Drużkowski, L. M. "The Jacobian Conjecture." IMPAN Preprint 492. Kraków, Poland: Math. Inst. Jagiellonian Uni...