{"slug": "jacobian-conjecture-from-wolfram-mathworld", "title": "Jacobian Conjecture from Wolfram MathWorld", "summary": "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.", "body_md": "The Jacobian conjecture asserts that a polynomial map having a nonzero\nconstant Jacobian determinant is an automorphism.\nIn the plane, first stated by Keller (1939), it says that a ring map of (the polynomial ring in two variables over the complex\nnumbers) to itself that fixes and sends , to , , respectively, is an automorphism iff\nthe Jacobian\nis a nonzero element of . The condition is easily shown to be necessary. The plane\ncase remains open.\n\nThere 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.\n\nIn July 2026, Alpöge (2026) announced the following polynomial counterexample, which he credited to the AI system Fable. Writing , its coordinate polynomials are\n\nAbhyankar, S. S. Lectures on Expansion Techniques in Algebraic Geometry. Bombay, India: Tata Institute\nof Fundamental Research, 1977.Alpöge, L. X post, July 20, 2026.\nhttps://x.com/__alpoge__/status/2079028340955197566.Bass,\nH. \"Conjecture jacobienne et opérateurs différentiels.\" Mém.\nSoc. Math. France, No. 38, 39-50, 1989.Bass, H.; Connell, E. H.;\nand Wright, D. \"The Jacobian Conjecture: Reduction of Degree and Formal Expansion\nof the Inverse.\" Bull. Amer. Math. Soc.7, 287-330, 1982.Becker,\nT. and Weispfenning, V. Gröbner\nBases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag,\np. 330, 1993.Drużkowski, L. M. \"The Jacobian Conjecture.\"\nIMPAN Preprint 492. Kraków, Poland: Math. Inst. Jagiellonian University, 1991.Formanek,\nE. \"Observations About the Jacobian Conjecture.\" Houston J. Math.20,\n369-380, 1994.Hochster, M. \"Lectures on Jacobian Conjecture.\"\nsci.math.research post forwarded by I. Algol. Nov. 11, 2004.Keller,\nO.-H. \"Ganze Cremona Transformationen.\" Monatsh. für Math. u. Phys.47,\n299-306, 1939.Meisters, G. H. \"Jacobian Problems in Differential\nEquations and Algebraic Geometry.\" Rocky Mountain J. Math.12,\n679-705, 1982.Meisters, G. H. \"Wanted: A Bad Matrix.\"\nAmer. Math. Monthly102, 546-550, 1995.Smale, S. \"Mathematical\nProblems for the Next Century.\" Math. Intelligencer20, No. 2,\n7-15, 1998.Smale, S. \"Mathematical Problems for the Next Century.\"\nIn Mathematics:\nFrontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax,\nand B. Mazur). Providence, RI: Amer. Math. Soc., 2000.Zhang, Z.\n\"Direct Consequences of the Three-Dimensional Counterexample to the Jacobian\nConjecture.\" July 20, 2026. https://zzhang-iu.github.io/papers/direct-consequences-jacobian/.", "url": "https://wpnews.pro/news/jacobian-conjecture-from-wolfram-mathworld", "canonical_source": "https://mathworld.wolfram.com/JacobianConjecture.html", "published_at": "2026-07-20 21:32:27+00:00", "updated_at": "2026-07-20 21:54:17.240387+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["Jacobian conjecture", "Keller", "L. Alpöge", "Fable", "Carolyn Dean", "Hochster", "Wolfram MathWorld"], "alternates": {"html": "https://wpnews.pro/news/jacobian-conjecture-from-wolfram-mathworld", "markdown": "https://wpnews.pro/news/jacobian-conjecture-from-wolfram-mathworld.md", "text": "https://wpnews.pro/news/jacobian-conjecture-from-wolfram-mathworld.txt", "jsonld": "https://wpnews.pro/news/jacobian-conjecture-from-wolfram-mathworld.jsonld"}}