Oval

In mathematics, an oval in a projective plane of order q is a set of q+1 points, no three collinear. In other words, an oval is a (q+1,2)-arc. Ovals in the Desarguesian projective plane PG(2,q) for q odd are just the nonsingular conics. Ovals in PG(2,q) for q even have not yet been classified. Ovals may exist in non-Desarguesian planes, and even more abstract ovals are defined which cannot be embedded in any projective plane.

Odd q

In a 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 statistical design of experiments.

Due to Segre's theorem (Segre 1955), every oval in PG(2, q) with q odd, is projectively equivalent to a nonsingular conic in the plane.

This implies that, after a possible change of coordinates, every oval of PG(2, q) with q odd has the parametrization :

Even q

When q is even, the situation is completely different.

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.

Given an oval there is a unique tangent through each point, and if q is even Qvist (1952) showed that all these tangents are concurrent in a point p outside the oval. Adding this point (called the nucleus of the oval) to the oval gives a hyperoval. Conversely, removing one point from a hyperoval immediately gives an oval.

Every nonsingular conic in the projective plane, together with its nucleus, forms a hyperoval. For each of these sets, there is a system of coordinates such that the set is:

However, many other types of hyperovals of PG(2, q) can be found if q > 8. Hyperovals of PG(2, q) for q even have only been classified for q < 64 to date.

Abstract ovals

Following (Bue1966), an abstract oval, also called a B-oval, of order n is a pair where F is a set of n + 1 elements, called points, and is a set of involutions acting on F in a sharply quasi 2-transitive way, that is, for any two with for , there exists exactly one with σ(a₁) = a₂ and σ(b₁) = b₂. Any oval embedded in a projective plane of order q might be endowed with a structure of an abstract oval of the same order. The converse is, in general, not true for ; indeed, for n = 8 there are two abstract ovals which may not be embedded in a projective plane, see (Fa1984).

When n is even, a similar construction yields abstract hyperovals, see (Po1997): an abstract hyperoval of order n is a pair where F is a set of n + 2 elements and is a set of fixed-point free involutions acting on F such that for any set of four distinct elements there is exactly one with σ(a) = b,σ(c) = d.

See also

• Ovoid (projective geometry)

References

• Buekenhout, F. (1966), "Études intrinsèque des ovales.", Rend. Mat. e Appl. 25 (5): 333-393, MR0218956 
• Faina, G. (1984), "Abstract hyperovals and Hadamard designs", J. Combin. Theory Ser. A 3: 307-314, MR0744079 
• Polster, B. (1997), "The B-ovals of order ", Australas. J. Combin. 16: 29-33, MR1477516 
• Qvist, B. (1952), "Some remarks concerning curves of the second degree in a finite plane", Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 1952 (134): 27, MR0054977 
• Segre, Beniamino (1955), "Ovals in a finite projective plane", Canadian Journal of Mathematics 7: 414–416, MR0071034, ISSN 0008-414X, http://www.cms.math.ca/cjm/v7/p414 

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