On Globally Diffeomorphic Polynomial Maps via Newton Polytopes and Circuit Numbers

TL;DR

This paper investigates conditions under which polynomial maps are globally diffeomorphic by analyzing Newton polytopes and circuit numbers, identifying a class where the Jacobian determinant's non-vanishing guarantees diffeomorphism, thus supporting a specific case of the Real Jacobian Conjecture.

Contribution

The paper introduces a novel approach using Newton polytopes and circuit numbers to characterize polynomial maps that are globally diffeomorphic when the Jacobian determinant never vanishes.

Findings

Identifies a class of polynomial maps where the Jacobian determinant's non-vanishing implies global diffeomorphism.

Provides conditions based on Newton polytopes at infinity for the global diffeomorphism property.

Supports a specific case of the Real Jacobian Conjecture for certain polynomial maps.

Abstract

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $\|F\|_2^2$. 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.