Vanishing ideals of projective spaces over finite fields and a projective footprint bound

TL;DR

This paper studies the vanishing ideal of projective spaces over finite fields, proving the generators form a universal Gr"obner basis, and introduces a projective footprint bound for estimating rational points.

Contribution

It establishes that known generators form a universal Gr"obner basis and develops a projective footprint bound for finite field varieties.

Findings

Generators form a universal Gr"obner basis

Developed a projective footprint bound

Applied to Serre's inequality for rational points

Abstract

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr\"obner basis of the ideal. Further we give a projective analogue of the footprint bound, and a version of it that is suitable for estimating the number of points of a projective algebraic variety over a finite field. An application to Serre's inequality for the number of rational points of projective hypersurfaces over finite fields is included