Georgia Institute of Technology College of Sciences

This dissertation consists of various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. We give a broad introduction and discuss connections to applied algebraic geometry, polyhedral, and tropical geometry. The individual topics discussed are as follows:

• Interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. Amongst other results, we characterize tropically convex sets in any dimension, and give a combinatorial description for the dimension of the tropical convex hull of an ordinary affine space. 
• The steady-state degree and mixed volume of a chemical reaction network. We present three case studies of infinite families of networks. For each family, we give a formula for the steady-state degree and mixed volume of the corresponding polynomial system. 
• Methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. 

Thesis may be viewed here.

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