A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates

TL;DR

This paper extends Pila-Wilkie type estimates for counting rational points on transcendental sets from characteristic zero to general local fields, advancing the understanding of rational point distribution in function field arithmetic.

Contribution

It generalizes existing rational point estimates to local fields of any characteristic, broadening their applicability in function field arithmetic.

Findings

Extended Pila-Wilkie estimates to local fields of any characteristic.

Unified rational point counting framework across different characteristics.

Bridged gap between characteristic zero and positive characteristic cases.

Abstract

We present here the natural extension of our Pila-Wilkie type estimates on the number of rational points of the trascendent part of a compact analytic subset of $\mathbb{F}_{q}((1/T))^{n}$ to analogous subsets of $K^{n}$, where $K$ is a general local field of any characteristic. That would integrate the analogous estimate provided by F. Loeser, G. Comte and R. Cluckers in characteristic 0.