The strong real Jacobian conjecture

Over ℂ a polynomial that never vanishes is constant, so “nowhere zero” and “constant” are the same hypothesis. Over ℝ they are not, and the weaker hypothesis gives a stronger conjecture. A constant nonzero determinant is in particular nowhere zero, so the strong real conjecture implies the real Jacobian question in every dimension.

It is false. Pinchuk's maps of ℝ² have everywhere positive Jacobian and are not injective, and padding with the identity carries the failure to every $n\ge2$. The conjecture holds for $n=1$, where a nowhere-zero derivative makes a polynomial strictly monotone.

Later work sharpened the example. Braun and Fernandes (2025) show that even a Jacobian bounded away from zero does not force injectivity. Braun, Gwoździewicz, Fernandes and Oréfice-Okamoto show that injectivity does follow when $\deg p\le6$, and give a non-injective example with $\deg p=7$ (unrefereed).^[1]

1. ↑ F. Braun, F. Fernandes, A counterexample to a folk real Jacobian conjecture, Israel J. Math. 267 (2025) 867–870. F. Braun, J. Gwoździewicz, F. Fernandes, B. Oréfice-Okamoto, A sharp degree bound in the real Jacobian conjecture, arXiv:2605.12302 (May 2026).