A method for constructing groups of permutation polynomials and its application to projective geometry

Abstract

This dissertation presents original work on permutation polynomials over finite fields. From a consideration of the proof of Cayley's theorem, it is clear that any finite group can be represented as a group of permutation polynomials (via interpolation) using the left regular action of the group on itself. The goal is to produce new families of permutation polynomials with particularly simple coefficients or relatively few terms. Central to the construction method is the choice of injective function from the group into a finite field, and a number of results are stated describing the relationship between the chosen injection and the resulting form of the representation polynomials. The construction method is then generalized to produce a single bivariate polynomial representing a given group, and several analogs of the univariate structure theorems are proved for the bivariate case. As applications of the method, we produce families of permutation polynomials representing various groups, several of which are new, and we use the bivariate representation to obtain new results on the construction of planar ternary rings coordinatizing finite Lenz-Barlotti type II.2 planes.