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
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