Thom isotopy theorem for non proper maps and computation of sets of stratified generalized critical values

Abstract

Let $X\subset {\mathbb{C}}^n$ be an affine variety and $f:X\to {\mathbb{C}}^m$ be the restriction to $X$ of a polynomial map ${\mathbb{C}}^n\to {\mathbb{C}}^m$. In this paper, we construct an affine Whitney stratification of $X$. The set $K(f)$ of stratified generalized critical values of $f$ can be also computed. We show that $K(f)$ is a nowhere dense subset of ${\mathbb{C}}^m$, which contains the set $B(f)$ of bifurcation values of $f$ by proving a version of the isotopy lemma for non-proper polynomial maps on singular varieties.