The weak Markus--Yamabe conjecture fails in dimension 14

Abstract

We show that the weak Markus--Yamabe conjecture fails in every dimension $n\ge 14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of ${\mathbb{R}}^n$ with component degrees $d_1,\dots ,d_n$ yields an explicit polynomial vector field on ${\mathbb{R}}^N$, with $N={\sum }_i\max (d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on ${\mathbb{R}}^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on ${\mathbb{R}}^{18}$. The dimensions $3\le n\le 13$ remain open.