Global multiplicity, special closure and non-degeneracy of gradient maps

Abstract

Given a polynomial map F:mathbb C^nlongrightarrow mathbb C^p with finite zero set, pgeqslant n, we introduce the notion of global multiplicity {text {m}}(F) associated to F, which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton non-degeneracy at infinity of F. This fact motivates us to study a combinatorial inequality concerning the normalized volume of global Newton polyhedra and to characterize the corresponding equality using special closures. We also study the Newton non-degeneracy at infinity of gradient maps.