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

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 ${\mathbb{𝔽}}_q\left[t\right]$-points of bounded degrees of algebraic varieties, uniformly in the cardinality $q$ of the finite field ${\mathbb{𝔽}}_q$ 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.