Slope-determinant method, complex cellular structures and hypersurface coverings of regular rational points

TL;DR

This paper advances the determinant method for estimating hypersurfaces covering rational points on varieties, incorporating complex cellular structures to improve bounds from subpolynomial to polylogarithmic factors.

Contribution

It introduces a novel combination of the determinant method with complex cellular structures, enhancing the bounds on hypersurface coverings of rational points.

Findings

Replaces subpolynomial bounds with polylogarithmic bounds

Integrates Arakelov geometry with cellular structures

Provides new estimates for covering rational points

Abstract

We use the determinant method of Bombieri-Pila and Heath-Brown and its Arakelov reformulation by Chen utilizing Bost's slope method to estimate the number of hypersurfaces required to cover the regular rational points with bounded Arakelov height on a projective variety. Using complex cellular structures introduced by Binyamini-Novikov, we replace the usual subpolynomial factor by a polylogarithmic factor in the estimation.