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Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 69 (35 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
POLYNOMIAL HOMOTOPIES FOR DENSE, SPARSE AND DETERMINANTAL SYSTEMS
, 1999
"... Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system ..."
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Cited by 12 (1 self)
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Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system
Homotopy Methods for Solving Polynomial Systems
, 2007
"... Homotopy continuation methods provide symbolicnumeric algorithms to solve polynomial systems. We apply Newton’s method to follow solution paths defined by a family of systems, a socalled homotopy. The homotopy connects the system we want to solve with an easier to solve system. While path followin ..."
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Cited by 2 (1 self)
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Homotopy continuation methods provide symbolicnumeric algorithms to solve polynomial systems. We apply Newton’s method to follow solution paths defined by a family of systems, a socalled homotopy. The homotopy connects the system we want to solve with an easier to solve system. While path following algorithms are numerical algorithms, the creation of the homotopy can be viewed as a symbolic operation. The performance of the homotopy continuation solver is primarily determined by its ability to exploit structure of the system. The focus of this tutorial is on linking recent algorithms
Homotopies for solving polynomial systems within a bounded Domain
 Theor. Comp. Sci
, 1994
"... The problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouch¶e's Theorem by homotopy continuation concepts yields a new class of homotopy methods, the socalled regional homotopy methods. These methods rely on isolating ..."
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Cited by 2 (0 self)
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The problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouch¶e's Theorem by homotopy continuation concepts yields a new class of homotopy methods, the socalled regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. As the dominant part has a sparser structure, it is easier to solve. It will be used as start system in the regional homotopy. The paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain. Applications illustrate the usefulness of the approach. 1