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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 76 (36 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.
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, ..."
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Cited by 71 (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 ...
Cylindrical algebraic subdecompositions
 MATHEMATICS IN COMPUTER SCIENCE
, 2014
"... Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semialgebraic sets. In this paper we introduce cylindrical algebraic subdecompositions (subCADs), which are subsets of CADs containing al ..."
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Cited by 3 (3 self)
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Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semialgebraic sets. In this paper we introduce cylindrical algebraic subdecompositions (subCADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of subCAD: variety subCADs which are those cells in a CAD lying on a designated variety; and layered subCADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truthtable invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.
Sampling Algebraic Sets in Local Intrinsic Coordinates
, 2009
"... Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descripti ..."
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Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.
A NOTE ON INVERSEORTHOGONAL TOEPLITZ MATRICES ∗
"... Abstract. In this note, inverseorthogonal Toeplitz matrices are investigated, and it is proved that every such a matrix is equivalent to a circulant one. As a corollary, it is showed that a real Hadamard matrix of order n> 2 with Toeplitz structure is necessarily circulant. ..."
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Abstract. In this note, inverseorthogonal Toeplitz matrices are investigated, and it is proved that every such a matrix is equivalent to a circulant one. As a corollary, it is showed that a real Hadamard matrix of order n> 2 with Toeplitz structure is necessarily circulant.