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

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.