On quadratic curves over finite fields

TL;DR

This paper explores the solution counts of quadratic curves, including conics with mixed terms, over finite fields, revealing diverse behaviors depending on the field's characteristic and degree.

Contribution

It extends previous work on algebraic curves over finite fields by analyzing quadratic conics with mixed terms and detailing their solution structures.

Findings

Solution counts vary significantly with field parameters.

Rich variety of solution behaviors for quadratic equations with mixed terms.

Provides new insights into the geometry of quadratic curves over finite fields.

Abstract

The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of solutions of the circle equation depends on the characteristic $p$ and the degree $n\geq 1$ of the finite field $\mathbb{F}_{p^n}$. In this paper, we make a similar study of the geometry over finite fields of the quadratic curves defined by the quadratic equations in two variables for the classical conic sections. In particular the quadratic equation with mixed term is interesting, and our results display a rich variety of possibilities for the number of solutions to this equation over a finite field.