On the classification of hyperovals

A hyperoval in the projective plane P2(Fq)P2(Fq) is a set of q+2q+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 P2(Fq)P2(Fq) are in one-to-one correspondence to polynomials with certain properties, called o-polynomials of FqFq. We classify o-polynomials of FqFq of degree less than 12q1/4. As a corollary we obtain a complete classification of exceptional o-polynomials, namely polynomials over FqFq that are o-polynomials of infinitely many extensions of FqFq.