Gale duality, decoupling, parameter homotopies, and monodromy
Date
2014
Authors
Niemerg, Matthew E., author
Bates, Daniel J., advisor
Shipman, Patrick, committee member
Peterson, Christopher, committee member
Lee, Chihoon, committee member
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Abstract
Numerical Algebraic Geometry (NAG) has recently seen significantly increased application among scientists and mathematicians as a tool that can be used to solve nonlinear systems of equations, particularly polynomial systems. With the many recent advances in the field, we can now routinely solve problems that could not have been solved even 10 years ago. We will give an introduction and overview of numerical algebraic geometry and homotopy continuation methods; discuss heuristics for preconditioning fewnomial systems, as well as provide a hybrid symbolic-numerical algorithm for computing the solutions of these types of polynomials and associated software called galeDuality; describe a software module of bertini named paramotopy that is scientific software specifically designed for large-scale parameter homotopy runs; give two examples that are parametric polynomial systems on which the aforementioned software is used; and finally describe two novel algorithms, decoupling and a heuristic that makes use of monodromy.
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Subject
gale duality
parameter homotopies
numerical algebraic geometry
monodromy