Key polynomials over valued fields

TL;DR

This paper investigates the structure of valuations on polynomial rings over valued fields, characterizing key polynomials and their associated graded algebras, and identifying valuations that lack key polynomials.

Contribution

It provides a detailed description of the set of key polynomials and the structure of the graded algebra for valuations on K[x], including criteria for valuations without key polynomials.

Findings

Characterization of the set of key polynomials for valuations on K[x]

Description of the graded algebra structure associated with these valuations

Identification of valuations that do not admit key polynomials

Abstract

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting key polynomials.