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

摘要

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 gk(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 gk.