Regeneration, local dimension, and applications in numerical algebraic geometry

Doctoral Dissertation

Abstract

Algorithms in the field of numerical algebraic geometry provide numerical methods for computing and manipulating solution sets of polynomial systems. One of the main algorithms in this field is the computation of the numerical irreducible decomposition. This algorithm has three main parts: computing a witness superset, filtering out the junk points to create a witness set, and decomposing the witness set into irreducible components. New and efficient algorithms are presented in this thesis to address the first two parts, namely regeneration and a local dimension test. Regeneration is an equation-by-equation solving method that can be used to efficiently compute a witness superset for a polynomial system. The local dimension test algorithm presented in this thesis is a numerical-symbolic method that can be used to compute the local dimension at an approximated solution to a polynomial system. This test is used to create an efficient algorithm that filters out the junk points. The algorithms presented in this thesis are applied to problems arising in kinematics and partial differential equations.

Attributes

Attribute NameValues
URN
  • etd-04142009-101147

Author Jonathan David Hauenstein
Advisor Andrew J. Sommese
Contributor Charles W. Wampler, Committee Member
Contributor Andrew J. Sommese, Committee Chair
Contributor Jason McLachlan, Committee Co-Chair
Contributor Bei Hu, Committee Member
Contributor Chris Peterson, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2009-03-27

Submission Date 2009-04-14
Country
  • United States of America

Subject
  • regeneration

  • polynomial systems

  • numerical algebraic geometry

  • local dimension

  • Bertini

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units

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