# Uniform approximation of Abhyankar valuation ideals in function field of prime characteristic

## TL;DR

This paper extends a key result on uniform approximation of valuation ideals from characteristic zero to prime characteristic using asymptotic test ideals, enhancing understanding in algebraic geometry over positive characteristic fields.

## Contribution

It provides the first prime characteristic analogue of a known characteristic zero theorem on valuation ideals, employing asymptotic test ideals.

## Key findings

- Established the prime characteristic version of the uniform approximation theorem
- Connected valuation ideals with asymptotic test ideals in positive characteristic
- Enhanced tools for algebraic geometry in prime characteristic fields

## Abstract

Using the theory of asymptotic test ideals, we prove the prime characteristic analogue of a characteristic $0$ result of Ein, Lazarsfeld and Smith (arXiv:math/0202303) on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties over perfect fields.