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A
Pila–Wilkie theorem for Hensel minimal curves
Victoria Cantoral Farfán, Kien Huu Nguyen, Mathias Stout and Floris Vermeulen |
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Vol. 3 (2024), No. 1, 119–145
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DOI: 10.2140/mt.2024.3.119
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Abstract |
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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
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Mathematical Subject Classification
Primary: 03C99, 14G05
Secondary: 03C65, 03C98, 11D88, 11G50
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Milestones
Received: 23 December 2022
Accepted: 5 March 2024
Published: 30 April 2024
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Authors |
| Victoria Cantoral Farfán | |
| Mathematisches Institut Georg-August Universität Göttingen Göttingen Germany |
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| 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 |
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| Floris Vermeulen | |
| Department of Mathematics KU Leuven Leuven Belgium |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). |
