Uniform lower bound of arithmetic Hilbert--Samuel function of hypersurfaces

TL;DR

This paper establishes an explicit, optimal lower bound for the arithmetic Hilbert-Samuel function of projective hypersurfaces and applies it to improve the determinant method.

Contribution

It provides a new uniform lower bound with the best possible dominant term for the arithmetic Hilbert-Samuel function of hypersurfaces.

Findings

Explicit lower bound with optimal dominant term

Application to the determinant method

Enhanced tools for arithmetic geometry

Abstract

In this article, we give an explicit and uniform lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which has the optimal dominant term. As an application, we apply this lower bound in the determinant method.