Number of points of a nonsingular hypersurface in an odd-dimensional projective space

TL;DR

This paper investigates the number of rational points on nonsingular hypersurfaces in odd-dimensional projective spaces over finite fields, providing an upper bound and classifying hypersurfaces that attain it.

Contribution

It establishes an upper bound for the number of points on such hypersurfaces and classifies all hypersurfaces that reach this bound, generalizing previous work on surfaces.

Findings

Upper bound for the number of points on nonsingular hypersurfaces

Complete classification of hypersurfaces attaining the bound

Extension of previous results from surfaces to higher dimensions

Abstract

The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space.