A characterisation of translation ovals in finite even order planes

TL;DR

This paper characterizes translation hyperovals in finite even order planes by linking combinatorial point sets in PG(4,q) to geometric structures in PG(2,q^2).

Contribution

It establishes a correspondence between certain combinatorial point sets in PG(4,q) and translation hyperovals in PG(2,q^2).

Findings

Points in PG(4,q) with specific properties correspond to translation hyperovals.

Existence of a regular spread in the hyperplane at infinity is shown.

The characterization is bidirectional, linking combinatorial and geometric structures.

Abstract

In this article we consider a set C of points in PG(4,q), q even, satisfying certain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the corresponding Bruck-Bose plane PG(2,q^2), the points corresponding to C form a translation hyperoval, and conversely.