Theory and Applications in Numerical Algebraic Geometry

Doctoral Dissertation

Abstract

Homotopy continuation techniques may be used to approximate all isolated solutions of a polynomial 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.

Attributes

Attribute NameValues
URN
  • etd-04102006-150155

Author Daniel James Bates
Advisor Andrew Sommese
Contributor Dave Severson, Committee Member
Contributor Charles W. Wampler II, Committee Member
Contributor Andrew Sommese, Committee Chair
Contributor Juan Migliore, Committee Member
Contributor Chris Peterson, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2006-04-07

Submission Date 2006-04-10
Country
  • United States of America

Subject
  • path tracking

  • numerical algebraic geometry

  • Bertini

  • homotopy continuation

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units

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