#### Thesis Title

#### Date of Graduation

Summer 2014

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Kishor Shah

#### 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

#### 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.

#### Copyright

© Amy Eva Fitzpatrick

#### Recommended Citation

Fitzpatrick, Amy Eva, "Groebner Bases" (2014). *MSU Graduate Theses*. 1655.

http://bearworks.missouristate.edu/theses/1655