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