# Counting the number of distinct distances of elements in valued field extensions

## TL;DR

This paper investigates the concept of distances in valued field extensions, establishing finiteness and bounds on the number of distinct distances, which aids in understanding defect extensions and resolution of singularities in positive characteristic.

## Contribution

It introduces a detailed study of distances in defect extensions, proving finiteness and providing bounds, with applications to valued function fields over perfect base fields.

## Key findings

- Number of distinct distances is finite in several cases.
- Upper bounds for the number of distances are computed.
- Results contribute to resolution of singularities in positive characteristic.

## Abstract

The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect extensions through the tool of distances, which measure how well an element in an immediate extension can be approximated by elements from the base field. We show that in several situations the number of essentially distinct distances in fixed extensions, or even just over a fixed base field, is finite, and we compute upper bounds. We apply this to the special case of valued functions fields over perfect base fields. This provides important information used in forthcoming research on relative resolution problems.