# Uniform Yomdin-Gromov parametrizations and points of bounded height in valued fields

## TL;DR

This paper establishes uniform parametrization results in non-Archimedean geometry, enabling bounds on algebraic points of bounded degree over finite fields and extending the Pila-Wilkie theorem to a uniform setting.

## Contribution

It introduces a uniform non-Archimedean Yomdin-Gromov parametrization framework applicable across varying finite fields and degrees, generalizing prior fixed-field results.

## Key findings

- Bound on the number of algebraic points of bounded degree in finite fields
- A uniform non-Archimedean Pila-Wilkie theorem established
- Extension of previous work to a more general, uniform setting

## Abstract

We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F_q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.