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

TL;DR

This paper develops a Whitney stratification for affine varieties and provides a method to compute the set of stratified generalized critical values of polynomial maps, extending Thom's isotopy theorem to non-proper maps on singular varieties.

Contribution

It introduces a way to compute stratified critical values and extends Thom's isotopy theorem to non-proper polynomial maps on singular varieties.

Findings

The set of stratified generalized critical values is nowhere dense in 

A version of the isotopy lemma is proved for non-proper polynomial maps

The set of bifurcation values is contained within the stratified critical values

Abstract

Let $X\subset\Bbb C^n$ be an affine variety and $f:X\to\Bbb C^m$ be the restriction to $X$ of a polynomial map $\Bbb C^n\to\Bbb C^m$. 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 $\Bbb C^m$, 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.