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

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 .