Jacobian conjecture

In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n {\displaystyle n} -dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse. The case n = 2 {\displaystyle n=2} (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved as of 2026. The case n = 1 {\displaystyle n=1} is trivially true, since the derivative of a polynomial is a nonzero constant only if the degree of the polynomial is 1 , {\displaystyle 1,} and linear polynomial functions are invertible. On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to the large language model Claude Fable 5, disproving the conjecture for ⁠ n > 2 {\displaystyle n>2} ⁠. The correctness of the counterexample is easy to verify with any computer algebra system. It has not been revealed, however, how it was found. Nevertheless, it led some mathematicians to elaborate on the mathematical reasons and the implications of the existence of the counterexample. For fields of positive characteristic, the usual Jacobian conjecture is false in one dimension, so the separable Jacobian conjecture, also called Adjamagbo's Jacobian conjecture, adds a separability premise to make the conjecture true in one dimension. The separable Jacobian conjecture has been proven false for all fields of positive characteristic in all dimensions n ≥ 2 {\displaystyle n\geq 2} , with the separable Jacobian conjecture for fields of characteristic zero coinciding with the usual Jacobian conjecture.