Non-monomial o-polynomials

The first infinite family of hyperovals not of the monomial type was discovered in 1985 by Payne [Pa85] in an investigation of a family of generalized quadrangles. He showed that for odd h,
f(x) = x1/6 + x3/6 + x5/6

is an o-polynomial (where the exponents are taken modulo 2h - 1). The collineation group stabilizing this hyperoval has order 2h and consists of an elation of order 2 and the automorphic collineations induced by the field automorphisms ([ThPaGe88]).

A second family of hyperovals (see [Ch88] and [Ch96]), again for odd h, is given by:

.   (22 mod(2h-1))

That this is an o-polynomial family was conjectured in 1985 and proved in 1995 ([Ch98]). In [OKPePr91], O'Keefe, Penttila and Praeger have shown that the full collineation stabilizer of this hyperoval is the cyclic group of field automorphisms acting in the plane as (x, y, z)(x,y,z) in the case of q = 32. In [OKTh96], O'Keefe and Thas have shown that any collineation stabilizing this point set and fixing either (0,1,0) or (1,0,0) must be one of these automorphic collineations for any q = 2h, h odd.

Using a connection between hyperovals and flocks of a quadratic cone([forward] Section 5), Cherowitzo, Pinneri, Penttila and Royle [ChPePiRo96], were able to produce an infinite family of non-monomial o-polynomials for all h (the only other such family is that of the monomial translation hyperovals). The hyperovals belonging to this family are called Subiaco hyperovals (named after a suburb of Perth, Australia, near the University of Western Australia). The Subiaco o-polynomial is given by:

f(x) = ((d2( x4 + x) + d2(1 + d + d2) (x3 + x2))(x4 + d2 x2 + 1)-1 + x½,

whenever tr(1/d) = 1 and d GF(4) if h 2 mod 4, where tr is the absolute trace function of GF(2h). This o-polynomial gives rise to a unique hyperoval if h2 mod 4 and to two inequivalent hyperovals if h 2 mod 4, h > 2 (see [PaPePi95]). The collineation stabilizer of the Subiaco hyperovals has been determined by Payne, Pinneri and Penttila [PaPePi95] and O'Keefe and Thas [OKTh96].

The most recent (1999) family consists of the Adelaide hyperovals [ChOKPe03] in planes of square order. As with the Subiaco family, they arise from a connection with flocks of a quadratic cone. The initial members of this family were found by computer search in the planes of order 64 and 256 (Penttila and Royle [PeRo95]). These were then shown to come from a recipe for finding cyclic q-clans ([forward] Section 5) developed by Payne. The recipe was used to obtain examples in PG(2, 1024) and PG(2, 4096). This work is reported in Payne, Penttila and Royle [PaPeRo97]. The description and proof of the infinite family in planes of square order is in Cherowitzo, O'Keefe and Penttila [ChOKPe03].

To describe the Adelaide hyperovals, we will start in a slightly more general setting. Let F = GF(q) and K = GF(q2). Let b K be an element of norm 1, different from 1, i.e. bq+1 = 1, b1. Consider the polynomial, for t F,

f(t) = (tr(b))-1tr(bm)(t + 1) + (tr(b))-1tr((bt + bq)m)(t + tr(b)t½+ 1)1-m + t½,

where tr(x) = trK/F(x) = x + xq. When h is even and m = ±(q - 1)/3, the above f(t) is an o-polynomial for the Adelaide hyperoval. It should be noted that for different choices of m, the function f(t) gives different hyperovals. For instance, if m = ±1 we get a hyperconic, if m = ±½ and h odd, we obtain a translation hyperoval (equivalent to t4) and if m = ±5 we obtain the Subiaco hyperoval (with some restrictions on b). It is not presently known if other choices of m will give hyperovals in any fields. Also, there is no formal proof that the Adelaide hyperovals are always distinct from the Subiaco hyperovals which exist in the same planes.

The only remaining example is a "sporadic" one in PG(2,32) obtained by O'Keefe and Penttila and will be described in the next section as we survey what is known about hyperovals in planes of small order.