Points below a parabola in affine planes of prime order

2024/11/28 by Sam Adriaensen, Adriaensen, Sam, Zsuzsa Weiner +1
Computer Science · Mathematics · #51E15 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications #Matrix Theory and Algorithms #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.19202

openalex publication_date 2024/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the set of points below the diagonal line. In this paper, we investigate the set of points lying below a parabola. We prove that in some sense, this set of points looks the same from all but two directions, despite having only one non-trivial automorphism. In addition, we study the sizes of these sets, and their intersection numbers with respect to lines.

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