Jacobian Conjecture from Wolfram MathWorld

Mathematician L. Alpöge announced a polynomial counterexample to the Jacobian conjecture on July 20, 2026, crediting the AI system Fable. The Jacobian conjecture, first stated by Keller in 1939, asserts that a polynomial map with a nonzero constant Jacobian determinant is an automorphism, and the plane case has remained open despite numerous incorrect proofs.

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 Abhyankar, 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 University, 1991.Formanek, E. "Observations About the Jacobian Conjecture." Houston J. Math.20, 369-380, 1994.Hochster, M. "Lectures on Jacobian Conjecture." sci.math.research post forwarded by I. Algol. Nov. 11, 2004.Keller, O.-H. "Ganze Cremona Transformationen." Monatsh. für Math. u. Phys.47, 299-306, 1939.Meisters, G. H. "Jacobian Problems in Differential Equations and Algebraic Geometry." Rocky Mountain J. Math.12, 679-705, 1982.Meisters, G. H. "Wanted: A Bad Matrix." Amer. Math. Monthly102, 546-550, 1995.Smale, S. "Mathematical Problems for the Next Century." Math. Intelligencer20, No. 2, 7-15, 1998.Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur . Providence, RI: Amer. Math. Soc., 2000.Zhang, Z. "Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture." July 20, 2026. https://zzhang-iu.github.io/papers/direct-consequences-jacobian/.