Monomial hyperovals in Desarguesian planes
by Vis, Timothy L., Ph.D., UNIVERSITY OF COLORADO AT DENVER, 2010, 189 pages; 3441286
 Abstract: In a finite projective plane of order n, an oval is a set of n + 1 points such that no three of the points are collinear. Each point of an oval lies on a unique line tangent to the oval. When n is even the tangent lines to an oval meet in a common point with which the oval can be extended to form a hyperoval. In the plane PG (2, q) constructed from GF (q)3, irreducible conics provide examples of ovals. Segre [32, 33] showed that when q is odd, these are the only examples. The situation is quite different when q is even: numerous families and examples exist and classification seems intractable. The work of Glynn [11], however, provides a powerful tool for classifying monomial hyperovals (those described by monomials), suggesting that this may be a tractable problem. Every known monomial hyperoval is equivalent to one described with an exponent whose binary expansion has three or fewer nonzero bits. Work of Segre [34] and of Cherowitzo and Storme [5] has fully classified the monomial hyperovals described by two or fewer bits. We extend this work by classifying the monomial hyperovals described by three bits. Following certain reduction arguments, we use Glynn's criterion [11] to analyze possible exponents. The possibilities are restricted to eleven broad cases, which are eliminated systematically through more narrowly focused application of Glynn's criterion. With this classification complete, we discuss several avenues of research with the goal of showing that any monomial hyperoval can be described with three or fewer bits. We discuss alternate forms of monomial hyperovals, giving all forms for the known monomial hyperovals and some of the alternate forms for arbitrary one, two, and three bit exponents. We also discuss two means of transforming the points of a projective plane that may be useful in simplifying monomial hyperovals by viewing them in relation to other well-studied structures.

 Adviser William E. Cherowitzo School UNIVERSITY OF COLORADO AT DENVER Source DAI/B 72-03, p. , Feb 2011 Source Type Dissertation Subjects Mathematics Publication Number 3441286
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