An Algorithmic Proof of Suslin′s Stability Theorem for Polynomial Rings (original) (raw)
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An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings
arXiv (Cornell University), 1994
Let k be a field. Then Gaussian elimination over k and the Euclidean division algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SL n (k) or SL n (k[x]), n ≥ 2, as a product of elementary matrices. Suslin's stability theorem states that the same is true for the multivariate polynomial ring SL n (k[x 1 ,. .. , x m ]) with n ≥ 3. As Gaussian elimination gives us an algorithmic way of finding an explicit factorization of the given matrix into elementary matrices over a field, we develop a similar algorithm over polynomial rings.
A New Algorithm for the Computation of Canonical Forms of Matrices over Fields
Journal of Symbolic Computation, 1997
A new algorithm is presented for the computation of canonical forms of matrices over fields. These are the Primary Rational, Rational, and Jordan canonical forms. The algorithm works by obtaining a decomposition of the vector space acted on by the given matrix into primary cyclic spaces (spaces whose minimal polynomials with respect to the matrix are powers of irreducible polynomials). An efficient implementation of the algorithm is incorporated in the Magma Computer Algebra System.
Extensions of Faddeev's algorithms to polynomial matrices
2009
Starting from algorithms introduced in [Ky M. Vu, An extension of the Faddeev's algorithms, in: Proceedings of the IEEE Multi-conference on Systems and Control on September 3-5th, 2008, San Antonio, TX] which are applicable to one-variable regular polynomial matrices, we introduce two dual extensions of the Faddeev's algorithm to one-variable rectangular or singular matrices. Corresponding algorithms for symbolic computing the Drazin and the Moore-Penrose inverse are introduced. These algorithms are alternative with respect to previous representations of the Moore-Penrose and the Drazin inverse of one-variable polynomial matrices based on the Leverrier-Faddeev's algorithm. Complexity analysis is performed. Algorithms are implemented in the symbolic computational package MATHEMATICA and illustrative test examples are presented.
Fast error-free algorithms for polynomial matrix computations
29th IEEE Conference on Decision and Control, 1990
Matrices of pol nomials over rings and fields provide a unifying framework $r many control system design problems. These include dynamic compensator design, infinite dimensional systems, controllers for nonlinear systems, and even controllers for discrete event s stems. An important obstacle for utilizing these owerful matiematical tools in practical applications has been &e non-availability of accurate and efficient algorithms to carry through the precise error-free computations required b these algebraic methods. In this paper we develop highly ekcient, error-free a1 orithms, for most of the important computations needed in %near systems over fields or rings. We show that the structure of the underlying rings and modules is critical in designing such algorithms.
Algorithms for computing triangular decomposition of polynomial systems
Journal of Symbolic Computation, 2012
We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.
Linear Algebra and Its Applications, 2006
Barnett's method through Bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way. Two different uses of this method in computer algebra are introduced here. Firstly, we describe an algorithm for parameterizing the greatest common divisor of several polynomials in K[x, y], being x a parameter taking values in an real field K. Secondly, we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following Corless et al. [R.M. Corless, P.M. Gianni, B.M. Trager, S. Watt, The singular value decomposition for polynomial systems, in: ACM International Symposium on Symbolic and Algebraic Computation, 1995, pp. 195-207]. Given a family of polynomials whose coefficients are imperfectly known, we describe an algorithm for computing their approximate greatest common divisor by using, as main tools, Barnett's method and singular value decomposition computations. Furthermore, we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients.
Journal of Computational and Applied Mathematics, 1996
This paper presents a new numerical algorithm for the computation of the greatest common divisor (GCD) of several polynomials, based on system-theoretic properties. The specific algorithm, characterizes the GCD as the output decoupling zero polynomial of an appropriate linear system associated with the given polynomial set. The computation of the GCD is thus reduced to specifying a nonzero entry of a vector forming the compound matrix of a matrix pencil directly produced from the associated linear system. A detailed description of the implementation of the algorithm is presented and analytical proofs of its stability are also developed. The MATLAB code of the algorithm is also described in the appendix. Pro(S) = [pl(S),...,pm(S)] t = [P0,Pl,"" ,Pd] ed(s) = Pined(s), where Pm~ ~m× (a+ 1), ea(s) = [1, s, ..., se] t. By GCD {~m,d} --~b(s) we shall denote the GCD of the set. {~d} denotes any set of polynomials of N[s] of maximal degree d' <~ d, d fixed. Notation 2. (1) Qp,n denotes the set of strictly increasing sequences of p integers (1 ~< p ~< n) chosen from 1, 2, ..., n. If c~, [3 ~ Qp,, we say that ~ precedes [3 (~ < [3), if there exists an integer t (1 ~< t ~< p) for which ~1 = [31, ..., c~t-t = [3t-1, c~t = [3t, where ~ = [3i denote the elements of ~, [3, respectively.
Gröbner Bases and Polynomial Equations
2016
Let S = k[x1, x2, . . . , xn] denote a polynomial ring over a field k where x1, x2, . . . , xn are indeterminates. A Gröbner basis is a set of polynomials in S which has several remarkable properties which enable us to carry out standard operations on ideals, rings and modules in an algorithmic way. Every set of polynomials in S can be transformed into a Gröbner basis. This process generalises three important algorithms: (1) Gauss elimination method for solving a system of linear equations, (2) Euclid’s algorithm for finding the greatest common divisor and (3) The simplex method of linear programming. One of the goals of these two lectures is to explain how to reduce the problem of solving a system of polynomial equations to a problem of finding eigenvalues of commuting matrices. We will introduce term orders first on the set of monomials in S and define the concept of Gróbner basis of an ideal. Term orders on monomials in k[x1, x2, . . . , xn] The set of monomials in the polynomial...
Algorithms for the Computing Determinants in Commutative Rings
arXiv (Cornell University), 2017
Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the best.
The F 4 -algorithm for Euclidean rings
Central European Journal of Mathematics
In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.