# Extensions of a valuation from $K$ to $K[x]$

## TL;DR

This paper explores how to extend valuations from a field to its polynomial ring, focusing on key polynomials, pseudo-convergent sequences, and minimal pairs, with implications for local uniformization in algebraic geometry.

## Contribution

It reviews and compares various approaches to valuation extension, introduces a recent version of key polynomials by Spivakovsky, and relates these objects to improve understanding of local uniformization.

## Key findings

- Spivakovsky's key polynomials relate explicitly to pseudo-convergent sequences.
- Examples illustrate properties of key polynomials, pseudo-convergent sequences, and minimal pairs.
- The approach aims to improve results on local uniformization in positive characteristic.

## Abstract

In this paper we give an introduction on how one can extend a valuation from a field $K$ to the polynomial ring $K[x]$ in one variable over $K$. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will discuss the objects that have been introduced to describe such extensions. We will focus on key polynomials, pseudo-convergent sequences and minimal pairs. Key polynomials have been introduced and used by various authors in different ways. We discuss these works and the relation between them. We also discuss a recent version of key polynomials developed by Spivakovsky. This version provides some advantages, that will be discussed in this paper. For instance, it allows us to relate key polynomials, in an explicit way, to pseudo-convergent sequences and minimal pairs. This paper also provides examples that ilustrate these objects and their properties. Our main goal when studying key polynomials is to obtain more accurate results on the problem of local uniformization. This problem, which is still open in positive characteristic, was the main topic of the paper of the author and Spivakovsky in the proceedings of ALaNT 3.