Mixed Newton numbers and isolated complete intersection singularities

Abstract

Let f:(ℂn, 0) → (ℂp, 0) be a complete intersection with an isolated singularity at the origin. We give a lower bound for the Milnor number of f in terms of the mixed multiplicities of a set of monomial ideals attached to the Newton polyhedra of the component functions of f. The Milnor number of f equals the bound that we give when f satisfies a condition that we define and that extends the notion of Newton non-degenerate function studied by Kouchnirenko. Our techniques are based on the notion of integral closure of submodules and its relation with Buchsbaum–Rim multiplicity and mixed multiplicities of a set of ideals.