An Explicit Characteristic-$2$ Counterexample to the Separable Jacobian Conjecture

Abstract

Let $k$ be a field of characteristic $2$. We exhibit an explicit polynomial endomorphism $F: \mathbb{A}_k^3\to\mathbb{A}_k^3$ whose Jacobian determinant is identically $1$, whose induced extension of rational function fields has degree $3$, and which is nevertheless noninjective. Since $2\neq 3$, this gives a counterexample to the usual Adjamagbo, or separable, formulation of the Jacobian conjecture in characteristic $2$. Stabilization yields analogous counterexamples in every dimension $n\geq 3$