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20
Approximate Factorization of Multivariate Polynomials via Differential Equations
 Manuscript
, 2004
"... The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors ove ..."
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Cited by 37 (9 self)
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The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no e#cient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10 3 ).
On Approximate Irreducibility of Polynomials in Several Variables
"... We study the problem of bounding a polynomial away from polynomials which are absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolerance on its coefficients. Using an absolute irreducibility criterion due ..."
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Cited by 19 (7 self)
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We study the problem of bounding a polynomial away from polynomials which are absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolerance on its coefficients. Using an absolute irreducibility criterion due to Ruppert, we are able to find useful separation bounds, in several norms, for bivariate polynomials. We also use Ruppert's criterion to derive new, more effective Noether forms for polynomials of arbitrarily many variables. These forms lead to small separation bounds for polynomials of arbitrarily many variables.
The approximate GCD of inexact polynomials part II: a multivariate algorithm
 In ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput. (New
, 2004
"... This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appears to be the first practical package with such capabilities. The most signific ..."
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Cited by 19 (0 self)
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This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appears to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and accuracy as demonstrated in the results of computational experiment. In addition, two variations of a squarefree factorization algorithm for multivariate polynomials are proposed as an application of the GCD algorithm.
Numerical Factorization of Multivariate Complex Polynomials
 Theoretical Comput. Sci
, 2003
"... One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. ..."
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Cited by 14 (4 self)
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One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment.
PHCmaple: A Maple interface to the numerical homotopy algorithms in PHCpack
 In QuocNam Tran, editor, Proceedings of the Tenth International Conference on Applications of Computer Algebra (ACA’2004
, 2004
"... Our Maple package PHCmaple provides a convenient interface to the functions of PHCpack, a collection of numeric algorithms for solving polynomial systems using polynomial homotopy continuation, which was recently extended with facilities to deal with positive dimensional solution sets. The interface ..."
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Cited by 12 (7 self)
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Our Maple package PHCmaple provides a convenient interface to the functions of PHCpack, a collection of numeric algorithms for solving polynomial systems using polynomial homotopy continuation, which was recently extended with facilities to deal with positive dimensional solution sets. The interface illustrates the benefits of linking computer algebra with numerical software. PHCmaple serves as a first step in a larger project to integrate a numerical solver in a computer algebra system.
Approximate Bivariate Factorization, a Geometric Viewpoint
, 2007
"... We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its d ..."
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Cited by 9 (1 self)
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We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.
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.
Continuations and monodromy on random riemann surfaces
 In SNC ’09: Proceedings of the 2009 conference on Symbolic numeric computation
, 2009
"... Our main motivation is to analyze and improve factorization algorithms for bivariate polynomials in C[x, y], which proceed by continuation methods. We consider a Riemann surface X defined by a polynomial f(x, y) of degree d, whose coefficients are choosen randomly. Hence we can supose that X is smoo ..."
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Cited by 4 (1 self)
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Our main motivation is to analyze and improve factorization algorithms for bivariate polynomials in C[x, y], which proceed by continuation methods. We consider a Riemann surface X defined by a polynomial f(x, y) of degree d, whose coefficients are choosen randomly. Hence we can supose that X is smooth, that the discriminant δ(x) of f has d(d −1) simple roots, ∆, that δ(0) = 0 i.e. the corresponding fiber has d distinct points {y1,..., yd}. When we lift a loop 0 ∈ γ ⊂ C − ∆ by a continuation method, we get d paths in X connecting {y1,..., yd}, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to the loops turning around each point of ∆. Multiplying families of “consecutive ” transpositions, we construct permutations then subgroups of the symmetric group. This allows us to establish and study experimentally some conjectures on the distribution of these transpositions then on transitivity of the generated subgroups. These results provide interesting insights on the structure of such Riemann surfaces (or their union) and eventually can be used to develop fast algorithms.
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