The Pila-Wilkie theorem for subanalytic families: a complex analytic approach

TL;DR

This paper offers a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets, replacing smooth parametrizations with Weierstrass division, providing a new approach in the field.

Contribution

It introduces a complex analytic method using Weierstrass division to prove the Pila-Wilkie theorem for subanalytic sets, offering an alternative to existing smooth parametrization techniques.

Findings

Provides a new complex analytic proof of the Pila-Wilkie theorem

Replaces $C^r$-smooth parametrizations with Weierstrass division

Enhances understanding of subanalytic sets in complex analysis

Abstract

We present a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^r$-smooth parametrizations by a variant of Weierstrass division.