A generalization of the Bombieri-Pila determinant method

TL;DR

This paper extends the Bombieri-Pila determinant method from affine plane curves to higher-dimensional varieties, providing a new proof of Heath-Brown's theorem using real-analytic techniques.

Contribution

It generalizes the determinant method to higher dimensions and offers a novel proof of Heath-Brown's theorem through real-analytic methods.

Findings

Generalization of the determinant method to higher-dimensional varieties

A new proof of Heath-Brown's 'Theorem 14'

Application of real-analytic considerations in the proof

Abstract

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.