Points below a parabola in affine planes of prime order

TL;DR

This paper explores the geometric properties of points lying below a parabola in affine planes of prime order, revealing symmetry and intersection characteristics despite limited automorphisms.

Contribution

It extends prior work on points below a line to parabolae, demonstrating symmetry properties and analyzing set sizes and line intersections in finite affine planes.

Findings

Set of points below a parabola exhibits near-symmetry from multiple directions.

The size of these point sets can be characterized precisely.

Intersection numbers with lines are computed and analyzed.

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.