Publication Date

Summer 2019

Degree Type

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

Advisor

Elizabeth Gross

Keywords

Chemical reaction networks, Hypergraph coloring, Monomial ideals

Subject Areas

Mathematics

Abstract

Under the assumption of mass-action kinetics, every chemical reaction network has an associated polynomial dynamical system. Rather than study the dynamics of this system, we shall study the ideal generated by the polynomials in the system called the steady state ideal. In this thesis, we will show that there is a combinatorial way to determine the existence of monomials in the steady state ideal using the underlying structure of the network. This allows us to prove that there is a combinatorial condition that is enough to guarantee the steady state ideal is monomial. We introduce three operations on chemical reaction networks that preserve the steady state ideals. We are interested in classifying the chemical reaction networks with monomial ideals. In this thesis, we shall characterize a class of networks whose steady state ideal is monomial using the combinatorics on the network. This work can be viewed as the first step in the systematic study of steady state ideals. While we were able to define ideal preserving operations, the existence of a complete characterization of networks with monomial ideals still remains an open question.

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