Theses and Dissertations

Date of Award

12-2023

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Brandt Kronholm

Second Advisor

Timothy Huber

Third Advisor

Josef Sifuentes

Abstract

We illustrate a method to prove the unimodality of Gaussian polynomials ${N+m \brack m}$ for $m = 5$ and $6$, building upon Dr. Brandt Kronholm's work, which proved unimodality for $m = 2,3,$ and $4$. Our approach involves viewing coefficients $p(n,m,N)$ of Gaussian polynomials $N+m \brack m$ based on how far away $n$ is from the central coefficient $p(\lfloor\frac{mN}{2}\rfloor,m,N)$ and then creating generating functions for those coefficients. We then take the difference of neighboring generating functions and change those generating functions into quasipolynomials to verify that their coefficients are non-negative. While the generalization of these generating functions for the coefficients of Gaussian polynomials and then proving unimodality using Dr. Kronholm's novel proof for all $m$ remains unsolved, a conjecture along with some curious identities emerge from this effort.

Comments

Copyright 2023 Paul Marsh. All Rights Reserved.

https://go.openathens.net/redirector/utrgv.edu?url=https://www.proquest.com/dissertations-theses/quasipolynomials-unimodality-gaussian-polynomials/docview/2928510712/se-2?accountid=7119

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