Electronic Theses and Dissertations

Date

2022

Document Type

Thesis

Degree Name

Master of Science

Department

Mathematical Sciences

Committee Chair

David Grynkiewicz

Committee Member

Mate Wierdl

Committee Member

Vladimir S Nikiforov

Abstract

In this Master’s Thesis, we showcase the use of an array of results collectively known as the polynomial method. First, we lay groundwork, giving some basic definitions, notation, and prerequisites. Then, we introduce three related theorems, the Combinatorial Nullstellensatz, Generalized Combinatorial Nullstellensatz, and Punctured Combinatorial Nullstellensatz, each concerning properties of multivariate polynomials with specific constraints on their degree. With these results proven, we then showcase some simple combinatorial and graph theoretic results which have very simple proofs making use of the Nullstellensatz theorems. In chapter 4 we give the proof of the q-Dyson theorem as published by Zeilberger and Bressoud. Finally, in chapters 5, 6 and 7 we introduce zero-sum theory, prove the Davenport constant of finite abelian p-groups without making use of group rings, and then state and prove the Chevalley-Warning theorem and use it to prove the Kemnitz conjecture on zero-sums in cyclic groups.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Open access

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