Determination of hyperovals by lines through a few points

2023/09/19 by Ding, Zhiguo, Zieve, Michael E.
#05B25 (Primary) #11T06 #51E20 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.10866

Abstract

If S is a set of q+2 points in P²(F_q) such that some point of S is not on any line containing two other points of S, then in suitable coordinates S has the form S_f:=(c:f(c):1) : c in F_q U (1:0:0),(0:1:0) for some f(X) in F_q[X]. Let T be a subset of S_f which contains the two infinite points and at least 3+log₃(q)/4 finite points. We show that if there is no line passing through a point of T and two other points of S_f, and deg(f)<=q^1/4, then no three points of S_f are collinear, so that S_f is a hyperoval. We also determine all f(X) with deg(f)<=q^1/4 for which S_f is a hyperoval, which strengthens a result that was proved by Caullery and Schmidt using entirely different methods.

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