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Counting zeros of polynomials over finite fields


Erickson, Daniel Edwin (1974) Counting zeros of polynomials over finite fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Q28M-M322.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • McEliece, Robert J. (advisor)
  • Dilworth, Robert P. (advisor)
Thesis Committee:
  • Unknown, Unknown
Defense Date:20 September 1973
Record Number:CaltechETD:etd-10132005-082129
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4061
Deposited By: Imported from ETD-db
Deposited On:14 Oct 2005
Last Modified:21 Dec 2019 01:51

Thesis Files

PDF (Erickson_de_1974.pdf) - Final Version
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