On quadratic curves over finite fields

arXiv1802.10486MaRDI QIDQ6298367

Andreas Aabrandt, Vagn Lundsgaard Hansen

Publication date: 28 February 2018

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 $ngeq1$ 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.

Mathematics Subject Classification ID

Quadratic and bilinear Diophantine equations (11D09) Counting solutions of Diophantine equations (11D45) Curves over finite and local fields (11G20) Finite ground fields in algebraic geometry (14G15) Congruences; primitive roots; residue systems (11A07)

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