On two upper bounds for hypersurfaces involving a Thas' invariant

TL;DR

This paper classifies hypersurfaces over finite fields that attain two specific upper bounds for the number of rational points, involving Thas' invariant, providing a clearer understanding of their structure.

Contribution

It offers a classification of hypersurfaces reaching two elementary bounds related to Thas' invariant, up to projective equivalence.

Findings

Hypersurfaces reaching the bounds are classified.

The bounds involve Thas' invariant.

Results apply to hypersurfaces over finite fields.

Abstract

Let $X^n$ be a hypersurface in $\mathbb{P}^{n+1}$ with $n\geq 1$ defined over a finite field $\mathbb{F}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of $\mathbb{F}_q$-points on $X^n$ which involve a Thas' invariant.