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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 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...
Using monodromy to decompose solution sets of polynomial systems into irreducible components
 PROCEEDINGS OF A NATO CONFERENCE, FEBRUARY 25  MARCH 1, 2001, EILAT
, 2001
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Numerical Irreducible Decomposition using PHCpack
, 2003
"... Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the ..."
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Cited by 21 (14 self)
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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.
Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm. Accepted for publication
 in The International Journal of Computational Science and Engineering
"... Abstract: Our problem is to decompose a positive dimensional solution set of a polynomial system into irreducible components. This solution set is represented by a witness set, obtained by intersecting the set with random linear slices of complementary dimension. Points on the same irreducible compo ..."
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Cited by 6 (6 self)
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Abstract: Our problem is to decompose a positive dimensional solution set of a polynomial system into irreducible components. This solution set is represented by a witness set, obtained by intersecting the set with random linear slices of complementary dimension. Points on the same irreducible components are connected by path tracking techniques applying the idea of monodromy. The computation of a linear trace for each component certifies the decomposition. This decomposition method exhibits a good practical performance on solution sets of relatively high degrees defined by systems of low degree polynomials.
OPTIMAL AMBIGUITY FUNCTIONS AND WEIL’S EXPONENTIAL SUM BOUND
"... Abstract. Complexvalued periodic sequences, u, constructed by Göran Björck, are analyzed with regard to the behavior of their discrete periodic narrowband ambiguity functions Ap(u). The Björck sequences, which are defined on Z/pZ for p> 2 prime, are unimodular and have zero autocorrelation on (Z/p ..."
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Cited by 1 (0 self)
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Abstract. Complexvalued periodic sequences, u, constructed by Göran Björck, are analyzed with regard to the behavior of their discrete periodic narrowband ambiguity functions Ap(u). The Björck sequences, which are defined on Z/pZ for p> 2 prime, are unimodular and have zero autocorrelation on (Z/pZ) � {0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is Ap(u)  ≤ 2 / √ p + 4/p outside of (0, 0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power of Weil’s exponential sum bound, which, in turn, is a consequence of his proof of the Riemann hypothesis for finite fields. Such bounds are not only of mathematical interest, but they have direct applications as sequences in communications and radar, as well as when the sequences are used as coefficients of phasecoded waveforms. 1.
New Results on the Parameterisation of Complex Hadamard Matrices
, 2008
"... In this paper we provide an analytical procedure which leads to a system of (n − 2) 2 polynomial equations whose solutions give the parameterisation of the complex n × n Hadamard matrices. It is shown that in general the Hadamard matrices depend on a number of arbitrary phases and a lower bound for ..."
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In this paper we provide an analytical procedure which leads to a system of (n − 2) 2 polynomial equations whose solutions give the parameterisation of the complex n × n Hadamard matrices. It is shown that in general the Hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given. The moduli equations define interesting geometrical objects whose study will shed light on the parameterisation of Hadamard matrices, as well as on some interesting geometrical varieties defined by them.
New Results on the Parametrisation of Complex Hadamard Matrices
, 2002
"... In this paper we provide an analytical procedure which leads to a system of (n−2) 2 polynomial equations whose solutions will give the parametrization of the complex n ×n Hadamard matrices. The key ingredient is a new factorization of unitary matrices in terms of n diagonal phase matrices interlaced ..."
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In this paper we provide an analytical procedure which leads to a system of (n−2) 2 polynomial equations whose solutions will give the parametrization of the complex n ×n Hadamard matrices. The key ingredient is a new factorization of unitary matrices in terms of n diagonal phase matrices interlaced with n−1 orthogonal matrices each one generated by a real vector. The moduli equations define interesting geometrical objects whose study will shed light not only on the parametrization of Hadamard matrices but also on the rationally connected varieties.