Newton non-degenerate foliations and blowing-ups

Abstract

A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies some non-degeneracy conditions, in terms of its Newton polyhedra system. These conditions are similar to the ones of Kouchnirenko and Oka for the case of functions. We introduce the concept of logarithmic reduction of singularities and we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.