Numbers of points of hypersurfaces without lines over finite fields

TL;DR

This paper establishes an upper bound on the number of points on hypersurfaces over finite fields that contain no lines, relating it to the hypersurface's dimension, degree, and the size of the finite field.

Contribution

It provides a new upper bound for the point count of hypersurfaces without lines over finite fields, linking geometric properties to finite field parameters.

Findings

Derived an explicit upper bound formula for point counts

Connected geometric properties with finite field characteristics

Enhanced understanding of hypersurfaces without lines over finite fields

Abstract

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.