Non-singular points on Hypersurfaces over $\mathbb{F}_{q}$

TL;DR

This paper surveys results on counting points on hypersurfaces over finite fields and explores conditions ensuring the existence of non-singular points based on the size of the finite field.

Contribution

It provides a comprehensive overview of point counting results and establishes new criteria for the existence of non-singular points depending on the field size.

Findings

Conditions for guaranteed non-singular points based on field size

Summary of known point counting results over finite fields

New bounds relating hypersurface properties to finite field cardinality

Abstract

We survey a number of results on the counting of points on hypersurfaces defined over finite fields. We also investigate when one can be guaranteed a non-singular point on a projective hypersurface and give a condition on the cardinality of the ambient field to achieve this.