# Simultaneous Monomialization

## TL;DR

This paper presents a new proof of the simultaneous embedded local uniformization theorem in zero characteristic, utilizing a novel approach called simultaneous monomialization, which advances the understanding of resolution of singularities.

## Contribution

It introduces a new proof technique for uniformization using simultaneous monomialization, extending existing methods in algebraic geometry.

## Key findings

- New proof of uniformization theorem in zero characteristic
- Development of the key elements theory for monomialization
- Enhanced understanding of resolution of singularities

## Abstract

We give a new proof of the simultaneous embedded local uniformization Theorem in zero characteristic for essentially of finite type rings and for quasi excellent rings. The results are a consequence of the simultaneaous monomialization presented here. The methods develop the key elements theory that is a more subtle notion than the notion of key polynomials.