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

Fernando Hernando, Gary McGuire

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 $g_k$.

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

Abstract

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

Paper Structure

This paper contains 11 sections, 23 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Singular Points
  4. Homogeneous Components and Intersection Multiplicity
  5. Type I
  6. Type II
  7. Type III
  8. Segre Revisited.
  9. The case $k\equiv 2\ \hbox{$($mod}4\hbox{$)$}$
  10. The case $g_k(x,y)$ irreducible over $\mathbb{F}_2$
  11. Case $g_k(x,y)$ not irreducible over $\mathbb{F}_2$

Key Result

Theorem 1.1

For any fixed even positive integer $k$, if $k\not=6$ and $k\not=2^i$ then the set $D(x^k)$ is a hyperoval in $PG(2,q)$ for at most a finite number of values of $q$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.1
  • Theorem 3.3: Bezout's Theorem
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 30 more