A Pila--Wilkie theorem for Hensel minimal curves

TL;DR

This paper extends the Pila--Wilkie theorem to transcendental curves within Hensel minimal structures, introducing a new point counting method based on residue field dimension to achieve optimal bounds.

Contribution

It develops a Pila--Wilkie type theorem for Hensel minimal curves and introduces a novel dimension counting approach for point counting in this setting.

Findings

Proves a Pila--Wilkie theorem for Hensel minimal structures.

Introduces a new point counting method related to residue field dimension.

Establishes optimal bounds for transcendental curves in this context.

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.