Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes

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The existence of certain monomial hyperovals D(x k ) in the finite Desarguesian projective plane PG(2, q), q even, is related to the existence of points on certain projective plane curves g k (x, y, z). Segre showed that some values of k (k = 6 and 2 i ) give rise to hyperovals in PG(2, q) for infinitely many q. Segre and Bartocci conjectured that these are the only values of k with this property. We prove this conjecture through the absolute irreducibility of the curves g k .