On a weak Jelonek's real Jacobian Conjecture in $\R^n$

TL;DR

This paper investigates a weaker form of Jelonek's real Jacobian Conjecture, providing conditions under which polynomial local diffeomorphisms are bijective, and extends results to semialgebraic maps.

Contribution

It proves a sufficiency condition for bijectivity related to the set of non-proper points and generalizes the result to semialgebraic local diffeomorphisms.

Findings

Established a sufficient condition for bijectivity based on the set of non-proper points.

Extended the bijectivity result from polynomial to semialgebraic local diffeomorphisms.

Provided insights into the structure of maps satisfying a weak version of Jelonek's conjecture.

Abstract

Let $Y:\R^n\to\R^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek's real Jacobian Conjecture states that if $\codim(S_Y)\geq2$, then $Y$ is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for bijectivity. Furthermore, we generalize our result on bijectivity to semialgebraic local diffeomorphisms.