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21
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 ..."
Abstract

Cited by 56 (26 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.
Numerical Homotopies to compute generic Points on positive dimensional Algebraic Sets
 Journal of Complexity
, 1999
"... Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the com ..."
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Cited by 50 (24 self)
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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...
Sparse Elimination and Applications in Kinematics
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 49 (11 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
Gröbner Bases Of Ideals Defined By Functionals With An Application To Ideals Of Projective Points
, 1996
"... In this paper we study Odimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morhpisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, genera ..."
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Cited by 36 (4 self)
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In this paper we study Odimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morhpisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, generalizing the BuchbergerMöller algorithm for computing a basis of an ideals vanishing at a set of points and the FGLM basis conversion algorithm. As an application
Polynomial gcd computations over towers of algebraic extensions
 In Proc. AAECC11
, 1995
"... ..."
Orthogonal Maximal Abelian *Subalgebras of the N×n Matrices and Cyclic NRoots
 Institut for Matematik, U. of Southern Denmark
, 1996
"... It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian subalgebras (MASA's) in the n \Theta nmatrices. The same result holds trivially for n = 2 and n = 3, but de la Harpe, Jones, Munemasa and Watatani have shown that, for every prime number n 7, there ..."
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Cited by 33 (3 self)
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It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian subalgebras (MASA's) in the n \Theta nmatrices. The same result holds trivially for n = 2 and n = 3, but de la Harpe, Jones, Munemasa and Watatani have shown that, for every prime number n 7, there are at least two nonisomorphic pairs of MASA's in the n \Theta n matrices. We draw connections to the research of Backelin, Bjorck and Froberg on cyclic nroots, and use their classification of cyclic 7roots to construct five nonisomorphic pairs of MASA's in the 7 \Theta 7 matrices. 1 1 Introduction Let A and B be two maximal abelian subalgebras (MASA's) of the algebra of complex n \Theta n matrices. A and B are orthogonal in the sense of Popa [16], i.e. A " B = C 1 and of the product of the orthogonal projections EA and EB of M n (C ) onto A and B (with respect to the HilbertSchmidt norm) is equal to the orthogonal projection EA"B of M n (C ) onto C 1. This means that B ae M n (C ...
Verification of polynomial system solvers
 In Proceedings of AWFS 2007
, 2007
"... Abstract. We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which mani ..."
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Cited by 9 (8 self)
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Abstract. We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of wellknown polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve. Key words: Software verification, polynomial system solver, triangular decomposition. 1
Solving a System of Algebraic Equations with Symmetries
 Journal of Pure and Applied Algebra
, 1997
"... this article. ..."
Polynomial homotopies on multicore workstations. Accepted for publication
 in the proceedings of PASCO 2010
"... Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can com ..."
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Cited by 3 (3 self)
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Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can compute solutions more accurately in the same amount of time with threads, and thus achieve quality up. Focusing on polynomial evaluation and linear system solving (the key ingredients of Newton’s method) we can double the accuracy of the results with the quad doubles of QD2.3.9 in less than double the time, if we use all available eight cores on our workstation. 1