Quantitative properties of the non-properness set of a polynomial map

TL;DR

This paper investigates the structure of the non-properness set of polynomial maps, establishing bounds on the degrees of parametric curves covering these sets, with applications to the Jacobian Conjecture and fixed point sets of algebraic group actions.

Contribution

It proves the degree bounds for parametric curves covering the non-properness set of polynomial maps, extending to real algebraic sets and applications to fixed point properties.

Findings

The non-properness set is covered by parametric curves of degree at most d-1.

The bounds are optimal and extend to real algebraic sets.

Application to fixed points of unipotent group actions shows they have no isolated points.

Abstract

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric curves of degree at most $d-1$. This bound is best possible. Moreover, we prove that if $X\subset\mathbb{R}^n$ is a closed algebraic set covered by parametric curves, and $f: X\rightarrow\mathbb{R}^m$ is a generically finite polynomial map, then the set $S_f$ of non-properness of $f$ is also covered by parametric curves. Moreover, if $X$ is covered by parametric curves of degree at most $d_1$, and the map $f$ has degree $d_2$, then the set $S_f$ is covered by parametric curves of degree at most $2d_1d_2$. As an application of this result we show a real version of the Bia{\l}ynicki-Birula theorem: Let $G$ be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety $X\subset\mathbb{R}^n$. Then the set $Fix(G)$ of fixed points has no isolated points.