On quadratic curves over finite fields

2018/02/28 by Vagn Lundsgaard Hansen, Hansen, Vagn Lundsgaard, Andreas Aabrandt +1 · 1 citation
Computer Science · Mathematics · #11A07 #11D09 #11D45 #11G20 #14G15 #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1802.10486

openalex publication_date 2018/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 1 of the finite field \mathbbF_pⁿ. 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.

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