Newton Non-degenerate Foliations and Blowing-ups

TL;DR

This paper characterizes Newton non-degenerate foliations via their Newton polyhedra and proves their equivalence to having a specific type of logarithmic reduction of singularities, linking combinatorial and geometric properties.

Contribution

It establishes a precise equivalence between Newton non-degeneracy and the existence of a combinatorial logarithmic reduction of singularities for codimension one foliations.

Findings

Newton non-degenerate foliations satisfy Kouchnirenko and Oka conditions

Such foliations admit a combinatorial logarithmic reduction of singularities

The characterization links Newton polyhedra conditions to geometric resolution processes

Abstract

A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.