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

Abstract

[EN] Given a polynomial map F : Cn --> Cp with finite zero set, p (sic)n, we introduce the notion of global multiplicity 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.