On the classification of hyperovals

2014/03/12 by Florian Caullery, Caullery, Florian, Kai-Uwe Schmidt +1 · 1 citation
Mathematics · #05B25 #11T06 #51E20 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B25 #msc:11T06 #msc:51E20

paper · pdf · doi:10.48550/arxiv.1403.2880

arxiv created 2014/05/30 · arxiv updated 2014/06/02

Abstract

A hyperoval in the projective plane ℙ²(\mathbbF_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 ℙ²(\mathbbF_q) are in one-to-one correspondence to polynomials with certain properties, called o-polynomials of \mathbbF_q. We classify o-polynomials of \mathbbF_q of degree less than \frac12q^1/4. As a corollary we obtain a complete classification of exceptional o-polynomials, namely polynomials over \mathbbF_q that are o-polynomials of infinitely many extensions of \mathbbF_q.

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