Valuations on $K[x]$ approaching a fixed irreducible polynomial

TL;DR

This paper investigates valuations on polynomial rings approaching a fixed irreducible polynomial, characterizing their structure, maximum values of augmented valuations, and applications to Artin-Schreier extensions.

Contribution

It provides a new characterization of valuations in al V_F via graded rings and explicitly determines maximum augmented valuation values using Newton polygon slopes.

Findings

Characterization of valuations in al V_F via graded rings

Explicit maximum values of augmented valuations on key polynomials

Applications to Artin-Schreier extensions

Abstract

For a fixed irreducible polynomial $F$ we study the set $\mathcal V_F$ of all valuations on $K[x]$ bounded by valuations whose support is $(F)$. The first main result presents a characterization for valuations in $\mathcal V_F$ in terms of their graded rings. We also present a result which gives, for a fixed $\nu\in \mathcal V_F$ and a key polynomial $Q\in{\rm KP}(\nu)$, the maximum value that augmented valuations in $\mathcal V_F$ can assume on $Q$. This value is presented explicitly in terms of the slopes of the Newton polygon of $F$ with respect to $Q$. Finally, we present some results about Artin-Schreier extensions that illustrate the applications that we have in mind for the results in this paper.