AI Disproves Jacobian Conjecture with Counterexample

Key Facts

• The Jacobian Conjecture was proposed in 1939 by German mathematician Eduard Ott-Heinrich Keller.
• The conjecture states that for polynomial maps with a constant non-zero Jacobian determinant, a polynomial inverse must exist.
• On 20 July 2026, mathematician Levent Alpöge, affiliated with Harvard University and Anthropic, announced a disproof of the conjecture, assisted by the AI model Claude Fable 5.
• The counterexample involves a 216-character polynomial formula satisfying the constant determinant condition but lacking a polynomial inverse.
• The specific inputs (0, 0, 1/4), (1, -3/2), and (-1, 3/2) all map to the same output (1/4, 0, 0, 0), demonstrating failure of invertibility.
• The result was verified using symbolic computation software SymPy and the formal proof assistant Lean.
• Mathematicians including Jared Duker Lichtman of Stanford University examined and validated the findings.
• This problem is included in Stephen Smale's list of important unsolved problems for the 21st century.

Background & Context

The Jacobian Conjecture is a fundamental, long-standing open problem in algebraic geometry concerning the global invertibility of polynomial functions in several variables. It was first proposed by Eduard Ott-Heinrich Keller in 1939. The conjecture asks whether polynomial maps with a constant, non-zero Jacobian determinant must have polynomial inverses.

Despite significant efforts, the conjecture has remained unresolved for decades. The recent involvement of AI, specifically the Claude Fable 5 model developed by Anthropic and guided by Levent Alpöge, marks a novel approach enabling discovery of a counterexample that disproves the conjecture.

This development demonstrates a significant advancement in the collaboration between AI and human mathematicians in tackling deep theoretical questions by employing symbolic and formal verification tools.

Why This Matters for Exams

Competitive exams often include questions on prominent mathematical conjectures, their history, and recent developments, especially those highlighting interdisciplinary approaches involving AI and computer science. Understanding the Jacobian Conjecture, its 1939 origin, and its recent disproof in 2026 with AI assistance is essential for students seeking comprehensive knowledge in algebraic geometry and computational mathematics.

Familiarity with AI tools such as Claude Fable 5, SymPy, and Lean, and their roles in modern mathematical research, equips examinees to understand current trends where mathematics intersects with technology.

Points to Remember

• The Jacobian determinant refers to the determinant of the matrix of all first-order partial derivatives of a multivariable function.
• Eduard Ott-Heinrich Keller proposed the conjecture in 1939, which has been a major open problem in algebraic geometry.
• Levent Alpöge, affiliated with Harvard University and Anthropic, announced in July 2026 the disproof using AI assistance from Claude Fable 5.
• The counterexample involves polynomial mappings with inputs (0,0,1/4), (1,-3/2), and (-1,3/2) all mapping to the same output, violating the invertibility requirement.
• Verification methods included the computer algebra system SymPy and the formal proof assistant Lean.
• The conjecture is part of the notable list of unsolved problems prepared by Stephen Smale for the 21st century.
• The achievement highlights the growing impact of AI in solving complex mathematical challenges.