Polyhedral and tropical geometry in nonlinear algebra
Hill, Cvetelina Dimitrova
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This dissertation consists of four chapters on various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. Chapter one gives a broad introduction and discusses connections to applied algebraic geometry, polyhedral, and tropical geometry. Chapter two studies the interaction between tropical and classical convexity, with a focus on the tropical convex hull of convex sets and polyhedral complexes. We describe the tropical convex hull of a line segment and a ray. We show that tropical and ordinary convex hull commute in two dimensions, and we characterize tropically convex sets in any dimension. We show that the dimension of a tropically convex fan depends on the coordinates of its rays, and we give a combinatorial description for the dimension of the tropical convex hull of an ordinary affine space. Lastly, we prove a lower bound on the degree of a fan tropical curve using only tropical techniques. Chapter three studies the steady-state degree and mixed volume of a chemical reaction network. The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying polyhedral geometry associated with the corresponding polynomial system. We focus on three case studies of infinite families of networks. For each family, we give a formula for the steady-state degree and the mixed volume of the corresponding polynomial system. Chapter four presents 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. The algorithm we develop is implemented as a package in Macaulay2. To demonstrate our method, we provide several examples, including an example arising from chemical reaction networks.