Oval (projective plane)

In projective geometry an oval is a point set in a plane that is defined by incidence properties.

As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane.

The higher dimensional analog of an oval is an ovoid in a projective space.

For finite planes (i.e. the set of points is finite) we have a more convenient characterization:[2] A set of points in an affine plane satisfying the above definition is called an affine oval.

This statement can be verified by a straightforward calculation for any of the conics (such as the parabola or hyperbola).

Here are some results: For topological ovals the following simple criteria holds: An oval in a finite projective plane of order q is a (q + 1, 2)-arc, in other words, a set of q + 1 points, no three collinear.

Ovals in the Desarguesian (pappian) projective plane PG(2, q) for q odd are just the nonsingular conics.

In an arbitrary finite projective plane of odd order q, no sets with more points than q + 1, no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments.

In this case, sets of q + 2 points, no three of which collinear, may exist in a finite projective plane of order q and they are called hyperovals; these are maximal arcs of degree 2.

Conversely, removing any one point from a hyperoval immediately gives an oval.

There are only three small examples (in the Desarguesian planes) where the automorphism group of the hyperoval is transitive on its points[14] so, in general, there are different types of ovals contained in a single hyperoval.

Every nonsingular conic in the projective plane, together with its nucleus, forms a hyperoval.

The remaining points of the hyperoval (when h > 1) will have the form (t, f(t),1) where t ranges through the values of the finite field GF(2h) and f is a function on that field which represents a permutation and can be uniquely expressed as a polynomial of degree at most 2h - 2, i.e. it is a permutation polynomial.

b) To describe the Adelaide hyperovals, we will start in a slightly more general setting.

c) The Penttila-O'Keefe o-polynomial is given by: where η is a primitive root of GF(32) satisfying η5 = η2 + 1.

In (Lunelli & Sce 1958) the details of a computer search for complete arcs in small order planes carried out at the suggestion of B. Segre are given.

In 1975, M. Hall Jr. showed,[15] also with considerable aid from a computer, that there were only two classes of projectively inequivalent hyperovals in this plane, the hyperconics and the hyperovals found by Lunelli and Sce.

Korchmáros (1978) independently gave a constructive proof of this result and also showed that in Desarguesian planes, the Lunelli-Sce hyperoval is the unique irregular hyperoval (non-hyperconic) admitting a transitive automorphism group (and that the only hyperconics admitting such a group are those of orders 2 and 4).

O'Keefe & Penttila (1991) reproved Hall's classification result without the use of a computer.

Their argument consists of finding an upper bound on the number of o-polynomials defined over GF(16) and then, by examining the possible automorphism groups of hyperovals in this plane, showing that if a hyperoval other than the known ones existed in this plane then the upper bound would be exceeded.

Brown & Cherowitzo (2000) provides a group-theoretic construction of the Lunelli-Sce hyperoval as the union of orbits of the group generated by the elations of PGU(3,4) considered as a subgroup of PGL(3,16).

Also included in this paper is a discussion of some remarkable properties concerning the intersections of Lunelli-Sce hyperovals and hyperconics.

In Cherowitzo et al. (1996) it is shown that the Lunelli-Sce hyperoval is the first non-trivial member of theSubiaco family[16] In Cherowitzo, O'Keefe & Penttila (2003) it is shown to be the first non-trivial member of the Adelaide family.

[18] In O'Keefe, Penttila & Praeger (1991) the collineation groups stabilizing each of these hyperovals have been determined.

Note that in the original determination of the collineation group for the Payne hyperovals the case of q = 32 had to be treated separately and relied heavily on computer results.

Penttila & Royle (1994) cleverly structured an exhaustive computer search for all hyperovals in this plane.

This settled affirmatively a long open question of B. Segre who wanted to know if there were any hyperovals in this plane besides the hyperconics.

By refining the computer search program, Penttila & Royle (1994) extended the search to hyperovals admitting an automorphism of order 3, and found the hyperoval: which has an automorphism group of order 12 (η is a primitive element of GF(64) as above).

Penttila and Royle[20] have shown that any other hyperoval in this plane would have to have a trivial automorphism group.

This would mean that there would be many projectively equivalent copies of such a hyperoval, but general searches to date have found none, giving credence to the conjecture that there are no others in this plane.