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Type of Document Dissertation Author Lu, Ye Author's Email Address ylu4@nd.edu URN etd-07032006-155647 Title Finding all real solutions of polynomial systems Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title andrew sommese Committee Chair Keywords
- real solution
- polynormial systems
Date of Defense 2006-06-27 Availability unrestricted Abstract In this thesis we give numerical algorithms to find theone-dimensional and two-dimensional parts of the solution sets on
$R^N$ of systems
egin{equation}label{realSystem}
f(x):=left[egin{array}{c}
f_{1}(x_{1},ldots,x_{N}) \
vdots \
f_{n}(x_{1},ldots,x_{N})
end{array} ight]=0
end{equation}
of $n$ polynomials on $R^N$.
Typically, we want to find the solutions on $R^N$ as opposed to
the solutions on $C^N$ when we need to solve such a system of $n$
polynomials. However, the real solutions are much more complicated
and expensive to compute than the complex solutions. Our approach is
to find the real solutions starting with the known complex
components. Recently in cite{SVW1,SVW2,SVW3}, new techniques have
been successfully developed to numerically decompose complex
algebraic sets into irreducible components. With the help of this
decomposition and a Morse-theoretic decomposition, we give
algorithms for numerically computing the real solution sets. The
Morse-theoretic decomposition only works for multiplicity one
components. For the components of multiplicity at least two, we use
the technique of deflation to make them into reduced components in a
higher dimensional space. The one-dimensional and two-dimensional
real sets are the most interesting ones in applications. We focus on
these two cases in this thesis. An application of our algorithms to
mechanisms, specifically the Stewart-Gough platform robot, is
presented.
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