Minimal Pairs, Truncations and Diskoids

TL;DR

This paper explores the relationship between abstract key polynomials and minimal pairs, providing a geometric interpretation of valuations using diskoids, a generalization of classical balls in non-archimedean fields.

Contribution

It establishes a connection between key polynomials and minimal pairs and introduces diskoids for a geometric perspective on valuations.

Findings

Relation between valuations and minimal pairs clarified

Diskoids offer a new geometric interpretation of valuations

Extension of classical ball concepts to non-archimedean fields

Abstract

We build on the correspondence between abstract key polynomials and minimal pairs made by Novacoski and show how to relate the valuations that are generated by each object. We can then give a geometric interpretation of valuations built in this fashion. To do so we employ an object called diskoid, which is a generalisation of the classical concept of ball in non-archimedian valued fields.