A Note on Nullstellensatz over Finite Fields

TL;DR

This paper provides an expository overview of Nullstellensatz-like theorems over finite fields, discussing vanishing ideals and offering an alternative proof of Ore's inequality for affine hypersurfaces.

Contribution

It presents a clear exposition of Nullstellensatz results in finite fields and introduces a new proof of Ore's inequality.

Findings

Analysis of vanishing ideals over finite fields

Discussion of Nullstellensatz-like results in affine and projective spaces

Alternative proof of Ore's inequality

Abstract

We give an expository account of Nullstellensatz-like results when the base field is finite. In particular, we discuss the vanishing ideal of the affine space and of the projective space over a finite field. As an application, we include an alternative proof of Ore's inequality for the number of points of affine hypersurfaces over finite fields.