A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q -4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.
An oval, sometimes called a hyperoval, is a set of q + 2 points of PG(2, q), q = 2 h, such that no three are collinear. Since a k-arc is defined to be a set of k points, with no three being collinear, an oval is also a (q + 2)-arc. Such a set of points generalizes the properties of a non-degenerate conic plus its nucleus, through which all the tangents of the conic pass. It is an important problem of finite algebraic geometry to classify all the ovals of the plane up to the group of collineations of PG(2, q), which is generated by PGL(3, q) and the h field automorphisms. For the theory of ovals see [31, [51, or [101. Let the points of PG(2, q) be represented by homogeneous triples (i,j, k) over the finite field GF(q) in the usual way, and similarly let the lines be represented by dual coordinates [r, s, t]. Thus (i,j, k) is incident with [r, s, tl if and only if ir + js + kt = 0. Consider the diagram in Figure .