Jacobian Conjecture

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.

if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse

the modern version of the Jacobian conjecture in n dimensions was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients.

presented an explicit counterexample to the conjecture in three-dimensional space which he credited to Claude Fable 5 AI model.

The case n = 2 (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved as of 2026.