Truncated Local Uniformization of Formal Integrable Differential Forms

TL;DR

This paper establishes local uniformization for rational codimension one foliations and formal differential forms using a truncated approach that controls Newton polygons and avoids value accumulations.

Contribution

It introduces a truncated local uniformization method for integrable formal differential forms, enabling an inductive procedure in valuation theory.

Findings

Proves local uniformization for rational codimension one foliations in any dimension.

Develops a truncated approach to uniformization that simplifies the inductive process.

Provides techniques to control Newton polygons and prevent value accumulations.

Abstract

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential $1$-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.