AI Solves Jacobian Conjecture Problem

The Jacobian Conjecture, proposed in 1939 by German mathematician Eduard Ott-Heinrich Keller, is a long-standing problem in algebraic geometry and polynomial mappings. On 20 July 2026, mathematician Levent Alpöge, associated with Harvard University and Anthropic, announced that the AI model Claude Fable 5 helped disprove the conjecture by finding a counterexample.

Jacobian Conjecture

The Jacobian Conjecture concerns polynomial maps and asks whether a map with a non-zero constant Jacobian determinant must always have a polynomial inverse. In mathematical terms, the conjecture deals with global invertibility of polynomial functions in several variables.

Counterexample and Polynomial Map

The AI-assisted result produced a 216-character formula that met the required constant determinant condition but could not be reversed by a polynomial inverse. The counterexample included three distinct input coordinates, namely (0, 0, 1/4), (1, -3/2), and (-1, 3/2), which all mapped to the same output (1/4, 0, 0, 0).

Verification and Mathematical Context

The result was checked using symbolic and formal verification tools such as SymPy and Lean. It was also examined by mathematicians including Jared Duker Lichtman of Stanford University, and it relates to Stephen Smale’s list of unsolved problems for the 21st century.

Important Facts for Exams

• The Jacobian Conjecture was proposed in 1939 by Eduard Ott-Heinrich Keller.
• The conjecture belongs to algebraic geometry and polynomial mapping theory.
• SymPy is a computer algebra system, and Lean is a formal proof assistant.
• Stephen Smale included major unsolved problems in a list for the 21st century.

Claude Fable 5 is an AI model associated with Anthropic. Levent Alpöge is a mathematician linked with Harvard University and Anthropic.