Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Pila--Wilkie theorem for Hensel minimal curves

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

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.

A Pila--Wilkie theorem for Hensel minimal curves

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.

Paper Structure

This paper contains 19 sections, 17 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The Pila-Wilkie theorem
  3. Hensel minimality
  4. Counting dimension
  5. The counting dimension
  6. Counting in valued fields
  7. Main results
  8. Some basic properties
  9. Some examples
  10. Notation and background
  11. Hensel minimality
  12. Consequences of Hensel minimality
  13. $T_r$-approximation
  14. Counting rational points on transcendental and algebraic curves
  15. $T_r$-parametrizations
  16. ...and 4 more sections

Key Result

Theorem 2.2.1

Suppose that $K$ is a henselian valued field of equicharacteristic $0$ equipped with a 1-h-minimal structure. Fix a pseudo-uniformizer $t$ and a lift of the residue field $\tilde{k}$. Suppose that $\mathrm{acl}=\mathrm{dcl}$ in $K$ and that the subgroup of $b$-th powers in $k^\times$ has finite inde Furthermore, the constant $N$ can be taken to hold uniformly throughout all transcendental members

Theorems & Definitions (42)

  • Definition 2.1.1
  • Theorem 2.2.1
  • Theorem 2.2.2
  • Theorem 2.2.3
  • Proposition 2.3.1
  • proof
  • Example 2.4.1
  • Example 2.4.2
  • Example 2.4.3
  • Theorem 3.2.1: CHR, Jacobian property
  • ...and 32 more