Algorithms from the field of numerical algebraic geometry provide robust means to compute all isolated solutions of arbitrary systems of polynomials and to give a thorough numerical description of positivedimensional solution components. However, the number of isolated complex solutions tends to grow exponentially with respect to the number $N$ of equations and variables, limiting the size of tractable systems. This is a particular problem for systems arising from discretizations of ordinary differential equations because $N$ is directly related to mesh size, and finer meshes are generally more desirable. On the other hand, the number of interesting solutions is often far smaller than the total count, and much effort is wasted on complex solutions with no physical interpretation.In the first part of this thesis, a general method is given for constructing a homotopy that directly relates solutions on grids of different resolutions. This enables iterative generation of solutions on increasingly fine meshes, and makes it possible to filter out some nonphysical solutions to substantially reduce wasted computation time. In the second part, a new algorithm is given to compute the numerical irreducible decomposition of a general polynomial system over a finite algebraic extension of $Q$ by projecting the solution set onto a line and deducing the exact defining polynomial of the projection.
MeshExpanding Homotopies and Numerical Irreducible Decomposition Over a Number Field
Doctoral DissertationAbstract
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Author  Timothy M McCoy 
Advisor  Andrew Sommese 
Contributor  Bei Hu, Committee Member 
Contributor  Andrew Sommese, Committee Chair 
Contributor  Jonathan Hauenstein, Committee Member 
Contributor  Yongtao Zhang, Committee Member 
Degree Level  Doctoral Dissertation 
Degree Discipline  Applied and Computational Mathematics and Statistics 
Degree Name  PhD 
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Submission Date  20140715 
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Record Visibility  Public 
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