"Groebner Bases" by Amy Eva Fitzpatrick

Date of Graduation

Summer 2014

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Kishor Shah

Abstract

We study Groebner bases and their applications in our thesis. We give a detailed proof of Dickson's Lemma, closely following the proof in Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. We then provide a geometric proof of this lemma, in the case of two variables, using our own ideas. We use Buchberger's Algorithm to produce a Groebner basis given any set of generators for an ideal I. We also consider the problem of solving systems of polynomial equations using various methods. These methods include finding a reduced Groebner basis to form a modified system, computing the resultant to eliminate a variable, and using various commands in Wolfram Mathematica 9.0. We also show the relationships between elimination theory and Groebner bases.

Keywords

monomial order, Dickson's Lemma, polynomial division, S-polynomial, Groebner basis, Buchberger's criterion, Buchberger's Algorithm, elimination theory, resultant, Mathematica

Subject Categories

Mathematics

Copyright

© Amy Eva Fitzpatrick

Campus Only

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