#### Vol. 14, No. 6, 2020

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

### Raf Cluckers, Arthur Forey and François Loeser

Vol. 14 (2020), No. 6, 1423–1456
##### 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.

##### Keywords
rational points, points of bounded height, parametrizations
##### Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 03C98, 11D88, 11G50
##### Milestones
Revised: 9 January 2020
Accepted: 3 March 2020
Published: 30 July 2020
##### Authors
 Raf Cluckers University of Lille CNRS, UMR 8524 – Laboratoire Paul Painlevé F-59000 Lille France KU Leuven Department of Mathematics Leuven Belgium Arthur Forey D-Math ETH Zürich Zürich Switzerland François Loeser Institut Universitaire de France Sorbonne Université UMR 7586 CNRS Institut Mathématique de Jussieu Paris France