Dependent Artin-Schreier Defect Extensions and Strong Monomialization

TL;DR

This paper proves that a specific Artin-Schreier defect extension in a counter-example to strong monomialization is dependent, using valuation generating sequences to deepen understanding of valuation extensions.

Contribution

It establishes the dependence of a key Artin-Schreier defect extension in a notable counter-example, advancing the theory of valuation and monomialization.

Findings

The first Artin-Schreier defect extension in the counter-example is dependent.

Generating sequences of valuations are effective tools in analyzing defect extensions.

The result affirms a question posed by F.-V. Kuhlmann.

Abstract

In this paper we affirmatively answer a question posed by F.-V. Kuhlmann. We show that the first Artin-Schreier defect extension in Cutkosky and Piltant's counter-example to strong monomialization is a dependent extension. Our main tool is the use of generating sequences of valuations.