Keller's Jacobian conjecture

Let $\Bbbk$ be a field of characteristic zero, and let $F=(F_1,\dots,F_n)\colon \Bbbk^n\to\Bbbk^n$ be a polynomial map. Write

$$\operatorname{Jac} F=\det\bigl(\partial F_i/\partial x_j\bigr).$$

Keller's Jacobian conjecture. If $F$ has $\operatorname{Jac} F\in\Bbbk^{\times}$, then $F$ is invertible, with polynomial inverse.

The conjecture was open since 1939 in every dimension $n\ge2$, but was refuted in July 2026 by an explicit polynomial map $F\colon\mathbb{C}^3\to\mathbb{C}^3$ with constant Jacobian and no polynomial inverse.