A note on the Jacobian Conjecture

TL;DR

This paper investigates properties of polynomial mappings with non-vanishing Jacobian, showing conditions under which such mappings are surjective and analyzing the topological structure of their non-properness sets, contributing to the Jacobian Conjecture.

Contribution

It establishes new conditions relating the smoothness and connectedness of the non-properness set to the surjectivity of polynomial mappings with non-zero Jacobian.

Findings

If the non-properness set is smooth, the mapping is surjective.

The non-properness set cannot be connected, supporting the Nollet-Xavier Conjecture.

In two dimensions, the non-properness set cannot be a self-intersecting curve.

Abstract

Let $F:\Bbb C^n\to\Bbb C^n$ be a polynomial mapping with a non vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, the set $S_F$ can not be connected (this is the Nollet-Xavier Conjecture). Additionally, if $n=2$, then the set $S_F$ of non-properness of $F$ cannot be a curve without self-intersections.