A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set (original) (raw)

Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

SIAM Journal on Numerical Analysis, 2001

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of di erent 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 nding, 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 its multiplicity. The bound is sharp (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 zeroes of a nite number of polynomials.

On multiplicities in polynomial system solving

Transactions of the American Mathematical Society, 1996

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

SIAM Journal on Numerical Analysis, 2009

The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V , this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopy-based numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler.

An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations

1986

A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By "polynomial combination", the system is reduced to a special form which may be interpreted as a multiplication table for power products modulo the system. The zeros are then formed from an ordinary eigenvalue problem for the matrix of the multiplication table. Degenerate situations may be handled by perturbing them into general form and reaching the zeros of the unperturbed system via a homotopy method.

SINGULAR 2-2 – A Computer Algebra System for Polynomial Computations

Singular is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, alge-braic geometry, and singularity theory. Singular's main computational objects are polynomials, ideals and modules over a large variety of rings. Singular features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. The new, upcoming version 2-2 includes also algorithms for a wide class of non-commutative algebras (Plural) and the possiblity for dynamic extension of the program at run-time (dynamic modules). Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root– finding, visualisation, and many more related functionalities.

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Electron. Colloquium Comput. Complex., 2021

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [BM74], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans [Uma08] and Kedlaya & Umans [KU11] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables n is at most do(1) where the degree of the input polynomial in every variable is ...

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

Journal of Computational and Applied Mathematics, 2016

The contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects. Particularly, the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines. In this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance, a task that is seldom considered in literature. By the use of computer algebra system Mathematica, we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity. Applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations, we construct iterative methods that serve (i) for approximating a multiple zero of a given function f when the order of multiplicity is unknown and, simultaneously, (ii) for finding exact order of multiplicity. In particular, we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps. Such approach, combined with a rapidly convergent method for multiple zeros, provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy. The properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction.

Bounds on numers of vectors of multiplicities for polynomials which are easy to compute

Proceedings of the 2000 international symposium on Symbolic and algebraic computation symbolic and algebraic computation - ISSAC '00, 2000

Let F be an algebraically closed eld of zero characteristic, a polynomial ' 2 F X1; : : : ; Xn] have a multiplicative complexity r and f1; : : : ; fk 2 F X1; : : : ; Xn] be some polynomials of degrees not exceeding d, such that ' = f1 = = fk = 0 has a nite number of roots. We show that the number of possible distinct vectors of multiplicities of these roots is small when r; d and k are small. As technical tools we design algorithms which produce Gr obner bases and vectors of multiplicities of the roots for a parametric zerodimensional system. The complexities of these algorithms are singly exponential. We also describe an algorithm for parametric absolute factorization of multivariate polynomials. This algorithm has subexponential complexity in the case of a small (relative to the number of variables) degree of the polynomials.

Numerical factorization of multivariate complex polynomials

Theoretical Computer Science, 2004

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. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors. With a random combination of differentials we lower multiplicities and reduce to the regular case. Estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision. We apply our software to study the singularities of Stewart-Gough platforms.