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

## TL;DR

This paper characterizes translation hyperovals in projective spaces over finite fields using scattered linear sets of pseudoregulus type, extending previous results to higher dimensions and providing a new geometric perspective.

## Contribution

It introduces a new characterization of translation hyperovals via scattered linear sets of pseudoregulus type in higher-dimensional projective spaces.

## Key findings

- Affine point sets of translation hyperovals correspond to scattered linear sets of pseudoregulus type.
- The characterization generalizes previous results for PG(2,q^2).
- Provides a geometric framework linking hyperovals and linear sets.

## 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 Andr\'e/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$(2,q^2)$.