On the classification of hyperovals

Abstract

A hyperoval in the projective plane ${\mathbb{P}}^2({\mathbb{F}}_q)$ is a set of $q+2$ points no three of which are collinear. Hyperovals have been studied extensively since the 1950s with the ultimate goal of establishing a complete classification. It is well known that hyperovals in ${\mathbb{P}}^2({\mathbb{F}}_q)$ are in one-to-one correspondence to polynomials with certain properties, called o-polynomials of ${\mathbb{F}}_q$. We classify o-polynomials of ${\mathbb{F}}_q$ of degree less than $\frac{1}{2}{q}^{1/4}$. As a corollary we obtain a complete classification of exceptional o-polynomials, namely polynomials over ${\mathbb{F}}_q$ that are o-polynomials of infinitely many extensions of ${\mathbb{F}}_q$.