A note on the Jacobian Conjecture

Abstract

Let $F:{\mathbb{C}}^n\to {\mathbb{C}}^n$ 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.

Versions 4 and 5 are incorrect. There is an mistake in section 2. Versions 1,2,3 are correct. Sorry!