Surjectivity of linear operators and semialgebraic global diffeomorphisms

TL;DR

This paper establishes conditions under which certain semialgebraic local diffeomorphisms of Euclidean space are global diffeomorphisms, linking surjectivity of differential operators to global invertibility and proposing a new conjecture related to polynomial maps.

Contribution

It introduces a novel criterion involving surjectivity of linear partial differential operators for global diffeomorphism of semialgebraic maps and proposes a new analytic conjecture related to polynomial diffeomorphisms.

Findings

Surjectivity of specific differential operators implies global diffeomorphism.

A new conjecture for polynomial local diffeomorphisms is proposed.

Unification and generalization of previous results on global invertibility.

Abstract

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of $\mathbb{R}^n$ with non-properness set having codimension greater than or equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of $\mathbb{R}^n$. Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic mappings which turns out to unify and generalize previous results of the literature.