Improved Lang--Weil bounds for a geometrically irreducible hypersurface over a finite field

TL;DR

This paper improves bounds on the number of rational points on geometrically irreducible hypersurfaces over finite fields, using a probabilistic geometric approach to nearly achieve optimal estimates.

Contribution

It introduces a probabilistic Bertini-type method to sharpen existing bounds for rational points on hypersurfaces over finite fields.

Findings

Bounds are nearly optimal for large finite fields.

Probabilistic intersection techniques effectively estimate rational points.

Method improves upon previous explicit and asymptotic bounds.

Abstract

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we study the number of $\mathbb{F}_q$-points on the intersection of the given hypersurface with a random plane.