Approximation types describing extensions of valuations to rational function fields

TL;DR

This paper introduces the concept of approximation types to describe and classify extensions of valuations from a base field to its rational function field, providing a new framework for understanding valuation extensions.

Contribution

It defines approximation types for valuation extensions, establishes their uniqueness, and proves a bijective correspondence with extensions in certain algebraically closed or dense fields.

Findings

Approximation types uniquely describe valuation extensions.

They correspond bijectively to extensions when the base field is algebraically closed or dense.

The framework captures both immediate and non-immediate extensions.

Abstract

We introduce the notion of {\it approximation type} for the partial, and in certain cases the total description of extensions of a given valuation from a field $K$ to the rational function field $K(x)$. To every extension, a unique approximation type of $x$ over $K$ is associated, while $x$ may be the limit of many pseudo Cauchy sequences. Approximation types also provide information in cases where the extensions are not immediate, and we prove that they correspond bijectively to the extensions when $K$ is algebraically closed or lies dense in its algebraic closure.