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Monomial o-polynomials

A special class of hyperovals, containing most of the known examples, consists of those which are projectively equivalent to a hyperoval having a monomial o-polynomial. Such an o-polynomial must be of the form f(x) = xk. We define
(h) = {k | xk is an o-polynomial over GF(2h)}.
As has been observed by numerous authors ([Ch88],[Gl83],[Hi79],[Pa71]):

Result 1: If k (h) then 1/k, 1-k, 1/(1-k), k/(k-1) and (k - 1)/k (h) where these numbers are taken modulo 2h - 1. These six o--polynomials give projectively equivalent hyperovals.

We give a brief description of what is known to be in (h).

2 (h)h.
These are the hyperconics, the only family of monomial hyperovals to be found in all planes under consideration.

2i(h) if and only if (i,h)= 1.
These hyperovals were determined by Segre [Se57] in 1957. They are called translation hyperovals since they admit as an automorphism group a group of translations which is transitive on the affine points of the hyperoval. When i1, or h-1, these hyperovals are not equivalent to hyperconics. Payne [Pa71] has shown that these are the only additive o--polynomials. An examination of the Euler totient function reveals that when h = 5 or h 7, the translation hyperovals provide examples of hyperovals which are not complete conics (usually referred to as irregular hyperovals). The situation for h = 1,2,3 and 4 is well known, all hyperovals in these planes are hyperconics, but for h = 6 no examples other than the hyperconics were known. The question of whether or not irregular hyperovals exist in the Desarguesian plane of order 64, open since 1957, has finally been settled in the affirmative ([forward] Hyperovals in PG(2,64)).

6 (h) for h odd.
Discovered by Segre [Se62] in 1962, but most of the proofs appear in the 1971 treatment by Segre and Bartocci [SeBa71].

+ and 3+ 4 (h) for h odd, where and are automorphisms such that 422 mod (2h-1).
These two families were discovered by Glynn [Gl83] in 1982 (a few of the initial members of these families in small planes were already known). Alternate versions of the proofs that these are hyperovals can be found in [Ch98].

Glynn implemented a fast algorithm for determining membership in (h) and searched all values of h up to and including 19 as a prelude to the above-mentioned result. He has since extended this search and found no new hyperovals. We record this as:

Result 2 (Glynn [Gl89]): The sets (h) are completely determined for h28.

The collineation groups stabilizing the monomial o-polynomial hyperovals have been studied by O'Keefe and Penttila [OKPe94].

Another approach to classifying the monomial o-polynomials, inspired by work on flocks of cones (see below), is concerned with the number of non-zero bits in the binary representation of the exponent of the monomial. The one bit exponents correspond to the translation o-polynomials. The two bit exponents have been classified by Cherowitzo and Storme [ChSt98]. The three bit exponent classification is being worked on.


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