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.
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