Hyperovals in Desarguesian Planes: Definitions

In a finite projective plane of order n, an oval is a set of n+1 points, no three of which are collinear. In a coordinatized Desarguesian plane, a conic is the set of points whose coordinates satisfy a non-degenerate quadratic equation. While every conic is easily seen to be an oval, the converse, proved in 1955 by B.Segre [Se55] for Desarguesian planes of odd order, is a rather surprising result. Segre's result does not extend to Desarguesian planes of even order and the classification problem for ovals in these planes remains complex.

In the even order case, every oval can be uniquely extended to a set of n+2 points, no three of which are collinear. These sets are called hyperovals. The additional point, which is the common intersection of all the tangent lines to the oval, is called the nucleus of the oval (this point has also been called the knot and the strange point of the oval). A conic together with its nucleus is a hyperoval called a hyperconic [more] (also known as a regular hyperoval or complete conic).

We shall restrict ourselves to the Desarguesian planes defined over the Galois Field GF(2h), and coordinatized in the standard way. To avoid the trivial, and sometimes exceptional case of the Fano plane, we shall assume when necessary that h> 1. Any hyperoval in such a plane contains at least 4 points, no three of which are collinear, and we may assume (by the Fundamental Theorem of Projective Geometry) that a hyperoval passes through the four points, (1,0,0), (0,1,0), (0,0,1) and (1,1,1). If is a hyperoval, it is thus completely determined by its affine points and we define:

y = f(x) if and only if (x,y,1) is a point of .
It is easily seen that f(x) is a permutation polynomial which is called an o-polynomial. There are numerous results on the form of an o-polynomial (see [Ch88], and especially [OKPe91a]).