The density of integral points on hypersurfaces of degree at least four

TL;DR

This paper develops new bounds for the density of integer solutions to polynomial equations of degree four or higher, employing an advanced exponential sum method to improve understanding of integral points on hypersurfaces.

Contribution

It introduces an iterated Heath-Brown $q$-analogue of van der Corput's method to derive sharper bounds for solutions on high-degree hypersurfaces.

Findings

Established new bounds for integer solutions density.

Applied iterated exponential sum techniques.

Enhanced understanding of solutions on degree ≥4 hypersurfaces.

Abstract

Let $f$ be a polynomial of degree at least four with integer-valued coefficients. We establish new bounds for the density of integer solutions to the equation $f=0$, using an iterated version of Heath-Browns $q$-analogue of van der Corput's method of exponential sums.