Counting zeros of polynomials over finite fields

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The main results of this dissertation are described in the following theorem: Theorem 5.1 If P is a polynomial of degree r = s(q-1) + t, with 0 < t <= q - 1, in m variables over GF(q), and N(P) is the number of zeros of P, then: 1) N(P) > [...] implies that P is zero. 2) N(P) < [...] implies that N(P) [...] where [...] where (q-t+3) [...] ct [...] t - 1. Furthermore, there exists a polynomial Q in m variables over GF(q) of degree r such that N(Q) = [...]. In the parlance of Coding Theory 5.1 states: Theorem 5.1 The next-to-minimum weight of the rth order Generalized Reed-Muller Code of length [...] is (q-t)[...] + [...] where c, s, and t are defined above. Chapter 4 deals with blocking sets of order n in finite planes. An attempt is made to find the minimum size for such sets.