Counting rational points on transcendental curves in valued fields

TL;DR

This paper extends the Pila--Wilkie counting theorem to transcendental curves in arbitrary valued fields, providing upper bounds on rational points using parametrizations and the determinant method.

Contribution

It generalizes counting results from o-minimal structures to valued fields of mixed characteristic without relying on $r$-th power maps.

Findings

Upper bounds on rational points in valued fields

Extension of Pila--Wilkie theorem to new setting

Use of parametrizations and determinant method

Abstract

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.