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A Pila–Wilkie theorem for Hensel minimal curves

Victoria Cantoral Farfán, Kien Huu Nguyen, Mathias Stout and Floris Vermeulen

Vol. 3 (2024), No. 1, 119–145
DOI: 10.2140/mt.2024.3.119
Abstract

Recently, a new axiomatic framework for tameness in henselian valued fields was developed by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen and termed Hensel minimality. In this article we develop Diophantine applications of Hensel minimality. We prove a Pila–Wilkie type theorem for transcendental curves definable in Hensel minimal structures. In order to do so, we introduce a new notion of point counting in this context related to dimension counting over the residue field. We examine multiple classes of examples, showcasing the need for this new dimension counting, and prove that our bounds are optimal.

Keywords
Pila–Wilkie theorem, nonarchimedean geometry, analogues to o-minimality, tame geometry on valued fields, model theory of valued fields, point counting, rational points of bounded height, parametrizations
Mathematical Subject Classification
Primary: 03C99, 14G05
Secondary: 03C65, 03C98, 11D88, 11G50
Milestones
Received: 23 December 2022
Accepted: 5 March 2024
Published: 30 April 2024
Authors
Victoria Cantoral Farfán
Mathematisches Institut
Georg-August Universität Göttingen
Göttingen
Germany
Kien Huu Nguyen
Department of Mathematics
KU Leuven
Leuven
Belgium
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen Normandie
Caen
France
Mathias Stout
Department of Mathematics
KU Leuven
Leuven
Belgium
Floris Vermeulen
Department of Mathematics
KU Leuven
Leuven
Belgium