English

The weak Markus--Yamabe conjecture fails in dimension 14

Algebraic Geometry 2026-08-05 v1 Dynamical Systems

Abstract

We show that the weak Markus--Yamabe conjecture fails in every dimension n14n\geq14. We first prove a chain realization theorem: every polynomial Keller map F=\I+HF=\I+H of Rn\R^n with component degrees d1,,dnd_1,\ldots,d_n yields an explicit polynomial vector field on RN\R^N, with N=imax(di,2)nN=\sum_i\max(d_i,2)-n, whose Jacobian matrix has spectrum {1}\{-1\} at every point and whose singularities are in bijection with any prescribed fiber of FF. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on R14\R^{14} with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on R18\R^{18}. The dimensions 3n133\leq n\leq13 remain open.

Keywords

Cite

@article{arxiv.2608.05392,
  title  = {The weak Markus--Yamabe conjecture fails in dimension 14},
  author = {Álvaro Castañeda and Gerardo Honorato and Francisco Valenzuela-Henríquez},
  journal= {arXiv preprint arXiv:2608.05392},
  year   = {2026}
}