Proof of a Conjecture of Segre and Bartocci on Monomial Hyperovals in Projective Planes

TL;DR

This paper proves a conjecture by Segre and Bartocci that only specific exponents generate monomial hyperovals in finite projective planes, by demonstrating the absolute irreducibility of associated algebraic curves.

Contribution

It establishes the conjecture that only certain exponents produce hyperovals, using algebraic geometry techniques to analyze the irreducibility of related curves.

Findings

Confirmed the conjecture for all relevant exponents

Demonstrated the absolute irreducibility of the curves g_k

Connected hyperoval existence to algebraic curve properties

Abstract

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 $g_k$.