Properness of Polynomial Maps with Newton Polyhedra

Abstract

We discuss the notion of properness of a polynomial map $\varvec{f}:\mathbb {K}^m\rightarrow \mathbb {K}^n$, $\mathbb {K}=\mathbb {C}$ or $\mathbb {R}$, at a point of the target. We present a method to describe the set of non-proper points of $\varvec{f}$ with respect to Newton polyhedra of $\varvec{f}$. We obtain an explicit precise description of such a set of $\varvec{f}$ when $\varvec{f}$ satisfies certain condition (1.5). A relative version is also given in Sect. 3. Several tricks to describe the set of non-proper points of $\varvec{f}$ without the condition (1.5) is also given in Sect. 5.