Explicit uniform estimate of arithmetic Hilbert-Samuel function of hypersurfaces

TL;DR

This paper provides explicit uniform bounds for the arithmetic Hilbert-Samuel function of projective hypersurfaces and applies these bounds to estimate the density of rational points with small heights.

Contribution

It introduces explicit, uniform upper and lower bounds for the arithmetic Hilbert-Samuel function of hypersurfaces, with optimal dominant terms.

Findings

Established explicit bounds with optimal dominant terms

Applied bounds to estimate rational point density

Provided tools for further arithmetic geometry research

Abstract

In this paper, we will give an upper bound and a lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which are uniform and explicit. These two bounds have the optimal dominant terms. As an application, we use the lower bound to get an estimate of the density of rational points with small heights in a hypersurface.