Counting Zeros of Polynomials Over Finite Fields

Author: Erickson, Daniel Edwin

Year: 1974

Degree: Dissertation (Ph.D.)

Advisor: Dilworth, Robert P.

Committee Member: Unknown, Unknown

Option: Mathematics; Economics

DOI: 10.7907/Q28M-M322

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.

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