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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 65 (31 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 57 (26 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...
Polynomial interpolation in several variables
, 2000
"... This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique inter ..."
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Cited by 51 (5 self)
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This is a survey of the main results on multivariate polynomial interpolation in the last twentyfive years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.
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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HBases for Polynomial Interpolation and System Solving
, 1999
"... This paper is mainly addressed more to readers with background in Numerical Analysis than to those who are more familiar with Commutative Algebra. Therefore, we first explain at an elementary level the underlying common structure of Hbases and Grobner bases and show parallels between the two conce ..."
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Cited by 17 (5 self)
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This paper is mainly addressed more to readers with background in Numerical Analysis than to those who are more familiar with Commutative Algebra. Therefore, we first explain at an elementary level the underlying common structure of Hbases and Grobner bases and show parallels between the two concepts. The dimensions of the vector spaces I " P d are described using the Hilbert function in section 3. There we also present modules of syzygies, a tool for analyzing linear dependencies among polynomials, and their dimensions. In section 4 we summarize the known methods for computing Hbases and turn then, in section 5, to regular sequences. There we show in thm. 5.3, that n polynomials in n variables are already an Hbasis, if their homogeneous parts of maximal degree have no other zero in common than 0. In contrast to Grobner bases, which are under mild additional conditions uniquely determined by the given ideal and the term ordering, Hbases are not unique. We give in section 6 an exact description of the structure of Hbases for a given ideal. The last two sections are devoted to applications. First we show in section 7 that Stetter's Eigenmethod also works with Hbases in place of Grobner bases and then finally in section 8 how Hbases can be used to solve multivariate polynomial interpolation problems. H. M. Moller and T. Sauer / Hbases for interpolation and system solving 3 2. Hbases and Grobner bases Here and in the following sections we consider polynomials in n variables x 1 ; : : : ; x n with coefficients from a field K , say the field Q of rational numbers or the field R of real numbers. For short, we write P := K [x 1 ; : : : ; x n ] : Given a set F ae P, the set I := 8 ! : X f2F h f f j h f 2 P; only finitely many h f 6= 0 9 = ; is the i...
Computer algebra and algebraic geometry  achievements and perspectives
 J. SYMBOLIC COMPUT
, 2000
"... In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of example ..."
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Cited by 12 (1 self)
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In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of examples, the usefulness of computer algebra to mathematical research. Computer algebra itself is a highly diversified discipline with applications to various areas of mathematics; many of these may be found in numerous research papers, proceedings or textbooks (cf. Buchberger and Winkler, 1998; Cohen et al., 1999; Matzat et al., 1998; ISSAC, 1988–1998). Here, I concentrate mainly on Gröbner bases and leave aside many other topics of computer algebra (cf. Davenport et al., 1988; Von zur Gathen and Gerhard, 1999; Grabmeier et al., 2000). In particular, I do not mention (multivariate) polynomial factorization, another major and important tool in computational algebraic geometry. Gröbner bases were introduced originally by Buchberger as a computational tool for testing solvability of a system of polynomial equations, to count the number of solutions (with multiplicities) if this number is finite and, more algebraically, to compute in the quotient ring modulo the given polynomials. Since then, Gröbner bases have become the major computational tool, not only in algebraic geometry. The importance of Gröbner bases for mathematical research in algebraic geometry is obvious and nowadays their use needs hardly any justification. Indeed, chapters on Gröbner bases and Buchberger’s algorithm (Buchberger, 1965) have been incorporated in many new textbooks on algebraic geometry such as the books of Cox et al. (1992, 1998) or the recent books of Eisenbud (1995) and Vasconcelos (1998), not to mention textbooks which are devoted exclusively to Gröbner bases, such as Adams and Loustaunou (1994),
Computing Gröbner Bases by FGLM Techniques in a Noncommutative Setting
 J. SYMBOLIC COMPUT
, 2000
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Constructing Polyhedral Homotopies on GridofClusters
, 2009
"... Abstract. Polyhedral homotopy continuation is a promising approach to solving largescale polynomial systems. However, the method faces a computational bottleneck in constructing the homotopy functions because we need to find all mixed cells for polynomial systems. The search for mixed cells can be ..."
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Abstract. Polyhedral homotopy continuation is a promising approach to solving largescale polynomial systems. However, the method faces a computational bottleneck in constructing the homotopy functions because we need to find all mixed cells for polynomial systems. The search for mixed cells can be formulated as a vertex enumeration problem for polyhedra. Among the search methods for solving the problem efficiently, the dynamic enumeration method by Mizutani, Takeda and Kojima appears to have a speed advantage. However, no matter how we improve the efficiency of the search methods, we may not avoid the fact that the search cost rises rapidly as the size of the polynomial system grows. In this paper, we propose a parallel dynamic enumeration that can deal with mixed cell enumeration for largescale polynomial systems. The introduction of data redistribution technique in the parallel implementation enables us to enhance parallelization efficiency. Numerical results show that the parallel method can enumerate all mixed cells of polynomial systems whose scale exceeds the capabilities of other methods. Key words.