On Globally Diffeomorphic Polynomial Maps via Newton Polytopes and Circuit Numbers

Abstract

In this article we analyze the global diffeomorphism property of polynomial maps $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $\Vert F{\Vert }_2^2$. This allows us to identify a class of polynomial maps $F$ for which their global diffeomorphism property on ${\mathbb{R}}^n$ is equivalent to their Jacobian determinant $\text{det }JF$ vanishing nowhere on ${\mathbb{R}}^n$. In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.

27 pages