Properness of Polynomial Maps with Newton Polyhedra

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.