We will provide a short list of the major open problems in the theory of hyperovals in Desarguesian planes.
- Classify the hyperovals in PG(2, 2h)
- This remains the chief problem in the area. Open since 1955, it is considered to be a very difficult problem. We believe that this might possibly be accomplished within the next ten years. The other problems listed here may be considered to be subproblems of this one.
- Classify the monomial hyperovals.
- This should be a much simpler problem given that these hyperovals have large automorphism groups, however this does not seem to have lead to an easy classification.
- Classify the hyperovals that are associated with flocks of cones (members of a herd).
- The monomial hyperovals associated with flocks of quadratic cones have been classified by Penttila and Storme. Penttila believes that all hyperovals associated with quadratic cones are now known. All known hyperovals, with the single exception of Penttila-O'Keefe, are associated with flocks of cones over translation ovals (-cones).
- Do there exist any hyperovals with a trivial automorphism group?
- The current record holder is the Penttila-O'Keefe hyperoval in PG(2,32) with an automorphism group of size 3. The existence of rigid hyperovals (i.e. having trivial automorphism groups), would make the classification problem much harder. The nonexistence of such hyperovals would finish the classification in PG(2,64).
- Is there a family of hyperovals which includes the Penttila-O'Keefe hyperoval?
- My guess is yes, but it will be difficult to find.
- Is the Penttila-O'Keefe hyperoval in the herd of some type of flock?
- Again I would venture to say yes, but this is probably just wishful thinking. Brown, O'Keefe, Payne, Penttila and Royle in a 2007/08 paper
have shown that this hyperoval is not in a herd associated with a flock of an -cone.