Translation hyperovals and ${\mathbb{F}}_2$-linear sets of pseudoregulus type

Abstract

In this paper, we study translation hyperovals in PG$(2,q^k)$. The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG$(2,q^k)$ are precisely those that have a scattered ${\mathbb{F}}_2$-linear set of pseudoregulus type in PG$(2k-1,q)$ as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG$(2,q^2)$.