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

Abstract

Let f be a generically finite polynomial map f: mathbb {C}^nrightarrow 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 Xsubset mathbb {R}^n is a closed algebraic set covered by parametric curves, and f: Xrightarrow 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 Białynicki-Birula theorem: Let G be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety Xsubset mathbb {R}^n. Then the set Fix(G) of fixed points has no isolated points.