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 by-products, 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.

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar 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...

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this paper we want to describe some recent developments in polynomial interpolation, especially those which lead to the construction Partially supported by DGES Spain, PB 96-0730 Partially supported by DGES Spain, PB 96-0730 and Programa Europa CAI-DGA, Zaragoza, Spain 2 M. Gasca and T. Sauer / Polynomial interpolation of the interpolating polynomial, rather than verification of its mere existence

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),

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 H--bases 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 H--bases 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 H--basis, 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, H--bases are not unique. We give in section 6 an exact description of the structure of H--bases 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 H--bases in place of Grobner bases and then finally in section 8 how H--bases can be used to solve multivariate polynomial interpolation problems. H. M. Moller and T. Sauer / H--bases for interpolation and system solving 3 2. H--bases 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...

this paper is to generalize FGLM algorithm to noncommutative polynomial rings

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