1,514 research outputs found
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Natural Density of Rectangular Unimodular Integer Matrices
In this paper, we compute the natural density of the set of k x n integer
matrices that can be extended to an invertible n x n matrix over the integers.
As a corollary, we find the density of rectangular matrices with Hermite normal
form [O Id]. Connections with Cesaro's Theorem on the density of coprime
integers and Quillen-Suslin's Theorem are also presented.Comment: 8 page
A generalization of Serre's conjecture and some related issues
AbstractSeveral topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining
A local construction of the Smith normal form of a matrix polynomial
We present an algorithm for computing a Smith form with multipliers of a
regular matrix polynomial over a field. This algorithm differs from previous
ones in that it computes a local Smith form for each irreducible factor in the
determinant separately and then combines them into a global Smith form, whereas
other algorithms apply a sequence of unimodular row and column operations to
the original matrix. The performance of the algorithm in exact arithmetic is
reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two
additional tests performe
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases
This paper is a sequel to "Computing diagonal form and Jacobson normal form
of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We
present a new fraction-free algorithm for the computation of a diagonal form of
a matrix over a certain non-commutative Euclidean domain over a computable
field with the help of Gr\"obner bases. This algorithm is formulated in a
general constructive framework of non-commutative Ore localizations of
-algebras (OLGAs). We split the computation of a normal form of a matrix
into the diagonalization and the normalization processes. Both of them can be
made fraction-free. For a matrix over an OLGA we provide a diagonalization
algorithm to compute and with fraction-free entries such that
holds and is diagonal. The fraction-free approach gives us more information
on the system of linear functional equations and its solutions, than the
classical setup of an operator algebra with rational functions coefficients. In
particular, one can handle distributional solutions together with, say,
meromorphic ones. We investigate Ore localizations of common operator algebras
over and use them in the unimodularity analysis of transformation
matrices . In turn, this allows to lift the isomorphism of modules over an
OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of
this lifting with the solutions of the original system of equations. Moreover,
we prove some new results concerning normal forms of matrices over non-simple
domains. Our implementation in the computer algebra system {\sc
Singular:Plural} follows the fraction-free strategy and shows impressive
performance, compared with methods which directly use fractions. Since we
experience moderate swell of coefficients and obtain simple transformation
matrices, the method we propose is well suited for solving nontrivial practical
problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio
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