# Extending valuations to the field of rational functions using pseudo-monotone sequences

## TL;DR

This paper generalizes Ostrowski's result on valuation extensions to the field of rational functions by using pseudo-monotone sequences, linking valuation rings to topological structures in algebraic extensions.

## Contribution

It introduces a framework using pseudo-monotone sequences to describe all valuation extensions to rational function fields, extending previous one-dimensional results.

## Key findings

- Valuation rings correspond to closed and open balls in the topology induced by $V$.
- Extensions are realized via pseudo-monotone sequences, generalizing pseudo-convergent sequences.
- The approach applies when the $V$-adic completion of $K$ is algebraically closed.

## Abstract

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly correspond, respectively, to the closed and the open balls of $K$ in the topology induced by $V$.