Ramification theory and key polynomials

TL;DR

This paper establishes a relationship between ramification groups and key polynomials in valued field extensions, providing formulas and conditions that enhance understanding of valuation theory and monomialization.

Contribution

It introduces a formula linking ramification group order to key polynomials and characterizes conditions for the absence of limit key polynomials in rank one valuations.

Findings

Ramification group order expressed via key polynomials and residue field characteristic

Condition for no limit key polynomials in rank one valuations

Monomialization theorem derived from ramification group conditions

Abstract

For a simple, normal and finite extension of a valued field, we prove that we can related the order of the ramification group of the field extension and the set of key polynomials associated to the extension of the valuation. More precisely, the order of this group can be expressed in terms of a product of a power of the characteristic of the residue field of the valuation and the effective degrees of the key polynomials. We also give a condition on the order of the ramification group so that there is no limit key polynomials for a valuation of rank one. This condition also allow us to have a monomialization theorem.