## 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** (also known as a
**regular hyperoval** or **complete conic**).

We shall restrict ourselves to the Desarguesian planes defined
over the Galois Field
**GF**(2^{h}), 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]).

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