Polynomial maps with maximal multiplicity and the special closure

Abstract

In this article we characterize the polynomial maps \(F:\mathbb {C}^n\rightarrow \mathbb {C}^n\) for which \(F^{-1}(0)\) is finite and their multiplicity \(\mu (F)\) is equal to \(n!\mathrm V_n(\widetilde{\Gamma }_{+}(F))\), where \(\widetilde{\Gamma }_{+}(F)\) is the global Newton polyhedron of F. As an application, we derive a characterization of those polynomial maps whose multiplicity is maximal with respect to a fixed Newton filtration.