Polyhedral and tropical geometry in nonlinear algebra

Hill, Cvetelina Dimitrova

Title:

Polyhedral and tropical geometry in nonlinear algebra

dc.contributor.advisor	Yu, Josephine
dc.contributor.author	Hill, Cvetelina Dimitrova
dc.contributor.committeeMember	Baker, Matthew
dc.contributor.committeeMember	Blekherman, Grigoriy
dc.contributor.committeeMember	Gross, Elizabeth
dc.contributor.committeeMember	Leykin, Anton
dc.contributor.department	Mathematics
dc.date.accessioned	2021-09-15T15:42:26Z
dc.date.available	2021-09-15T15:42:26Z
dc.date.created	2021-08
dc.date.issued	2021-07-26
dc.date.submitted	August 2021
dc.date.updated	2021-09-15T15:42:26Z
dc.description.abstract	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.
dc.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/65051
dc.language.iso	en_US
dc.publisher	Georgia Institute of Technology
dc.subject	nonlinear algebra
dc.subject	tropical geometry
dc.subject	polyhedral geometry
dc.subject	chemical reaction networks
dc.subject	monodromy
dc.subject	homotopy continuation
dc.subject	tropical convexity
dc.title	Polyhedral and tropical geometry in nonlinear algebra
dc.type	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.advisor	Yu, Josephine
local.contributor.corporatename	College of Sciences
local.contributor.corporatename	School of Mathematics
relation.isAdvisorOfPublication	0569eb14-6836-43ce-bbd4-195f87a50534
relation.isOrgUnitOfPublication	85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication	84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level	Doctoral