Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials (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.

A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set

Journal of Complexity, 2006

Let F1, F2, . . . , Ft be multivariate polynomials (with complex coefficients) in the variables z 1 , z 2 , . . . , z n . The common zero locus of these polynomials, V (F 1 , F 2 , . . . , F t ) = {p ∈ C n |F i (p) = 0 for 1 ≤ i ≤ t}, determines an algebraic set. This algebraic set decomposes into a union of simpler, irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. Multiplicity plays an important role in many practical applications. It determines roughly "how many times the component should be counted in a computation." Unfortunately, many numerical methods have difficulty in problems where the multiplicity of a component is greater than one. The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set. The method sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique.

A Deterministic PTAS for the Algebraic Rank of Bounded Degree Polynomials

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2019

We present a deterministic polynomial time approximation scheme (PTAS) for computing the algebraic rank of a set of bounded degree polynomials. The notion of algebraic rank naturally generalizes the notion of rank in linear algebra, i.e., instead of considering only the linear dependencies, we also consider higher degree algebraic dependencies among the input polynomials. More specifically, we give an algorithm that takes as input a set f := {f 1 ,. .. , f n } ⊂ F[x 1 ,. .. , x m ] of polynomials with degrees bounded by d, and a rational number > 0 and runs in time O((nmd) O(d 2) • M (n)), where M (n) is the time required to compute the rank of an n × n matrix (with field entries), and finally outputs a number r, such that r is at least (1 −) times the algebraic rank of f. Our key contribution is a new technique which allows us to achieve the higher degree generalization of the results by Bläser, Jindal, Pandey (CCC'17) who gave a deterministic PTAS for computing the rank of a matrix with homogeneous linear entries. It is known that a deterministic algorithm for exactly computing the rank in the linear case is already equivalent to the celebrated Polynomial Identity Testing (PIT) problem which itself would imply circuit complexity lower bounds (Kabanets, Impagliazzo, STOC'03). Such a higher degree generalization is already known to a much stronger extent in the noncommutative world, where the more general case in which the entries of the matrix are given by poly

A new algorithm for the geometric decomposition of a variety

Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99, 1999

In this article, we present a new method for computing the decomposition of a variety into irreducible components. It is based on a property of Bezoutian matrices, which allows us to compute a multiple of the Chow form of the isolated points of the variety and to deduce a rational representation of these points. This tools is used recursively to compute the irreducible components from the lowest to the highest dimension. The asymptotic complexity is of the same order than the best complexity bound known for this problem. Our approach provides a substantial simpli cation of the previous methods and yields bounds on the height of the polynomials involved in these representations. An implementation in maple of this algorithm is described at the end.

The singular value decomposition for polynomial systems

Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95, 1995

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard's u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coefficient case. We also briefly discuss an application of the modified Lazard's method to the location of singular points on approximately known projections of algebraic curves.

Numerical computation of minimal polynomial bases A generalized resultant approach.

We propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix F (s). The proposed method exploits the structure of the left null space of generalized Wolovich or Sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix. The entire procedure can be implemented using only orthogonal transformations of constant matrices and results to a minimal basis with orthonormal coefficients.

From algebraic sets to monomial linear bases by means of combinatorial algorithms

Discrete Mathematics, 1995

Let K be a field; let ~mK" be a finite set and let 3(N)mK[xl ..... x,] be the ideal of ~. A purely combinatorial algorithm to obtain a linear basis of the quotient algebra K Ix1 ..... x,]/3(~) is given. Such a basis is represented by an n-dimensional Ferrers diagram of monomials which is minimal with respect to the inverse lexicographical order ~<i.l.-It is also shown how this algorithm can be extended to the case in which ~ is an algebraic multiset. A few applications are stated (among them, how to determine a reduced Grfbner basis of 3(,~) with respect to %i.1. without using Buchberger's algorithm).

A new method for computing a column reduced polynomial matrix

Systems & Control Letters, 1988

A new algorithm is presented for computing a column reduced form of a given full column rank polynomial matrix. The method is based on reformulating the problem as a problem of constructing a minimal basis for the rigth nullspace of a polynomial matrix closely related to the original one. The latter problem can easily be solved in a numerically reliable way. Two examples illustrating the method are included.

Polynomial numerical hulls of matrices

Linear Algebra and its Applications, 2008

For any n-by-n complex matrix A, we use the joint numerical range W (A, A 2 , . . . , A k ) to study the polynomial numerical hull of order k of A, denoted by V k (A). We give an analytic description of V 2 (A) when A is normal. The result is then used to characterize those normal matrices A satisfying V 2 (A) = σ(A), and to show that a unitary matrix A satisfies V 2 (A) = σ(A) if and only if its eigenvalues lie in a semicircle, where σ(A) denotes the spectrum of A. When A = diag (1, w, . . . , w n−1 ) with w = e i2π/n , we determine V k (A) for k ∈ {2}∪{j ∈ N : j ≥ n/2}.