Dissertation Defense Announcement

 College of Arts and Sciences announces the Final Dissertation Defense of

Thomas Fleming

for the Degree of Master of Science

March 03, 2022 at 01:00 PM,Dunn Hall  Room 118

Advisor: David Grynkiewicz

Polynomial Methods: Recent Advancements in Combinatorics

ABSTRACT: "In this Master’s Thesis we showcase 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 constrains 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. In chapter 4 we give a more involved proof using the Combinatorial Nullstellensatz, the proof of the q-Dyson theorem as given by Zeilberger and Bressoud. Finally, in chapters 5, 6 and 7 we introduce zero-sum theory, prove a result on the Davenport constant of finite abelian p-groups, and then state and prove the Chevalley-Warning theorem and use it to prove Kemnitz conjecture on zero-sums in the product of two cyclic groups of order 2."