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Type of Document Dissertation Author Bates, Daniel James Author's Email Address dbates1@nd.edu URN etd-04102006-150155 Title Theory and Applications in Numerical Algebraic Geometry Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Andrew Sommese Committee Chair Charles W. Wampler II Committee Member Chris Peterson Committee Member Dave Severson Committee Member Juan Migliore Committee Member Keywords
- Bertini
- path tracking
- homotopy continuation
- numerical algebraic geometry
Date of Defense 2006-04-07 Availability unrestricted Abstract Homotopy continuation techniques may be used to approximate all isolated solutions of apolynomial system. More recent methods which form the crux of the young field known as
numerical algebraic geometry may be used to produce a description
of the complete solution set of a polynomial system, including the positive-dimensional
solution components. There are
four main topics in the present thesis: three novel numerical methods and one new
software package. The first algorithm is a way to increase precision as needed during
homotopy continuation path tracking in order to decrease the computational cost of
using high precision. The second technique is a new way to compute the scheme structure
(including the multiplicity and a bound on the Castelnuovo-Mumford regularity) of an ideal supported at a single
point. The third method is a new way to approximate all solutions of a certain class of
two-point boundary value problems based on homotopy continuation. Finally, the software package, Bertini, may be used
for many calculations in numerical algebraic geometry, including the three new algorithms
described above.
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