A new method for computing a column reduced polynomial matrix
Systems & Control Letters
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants
SIAM Journal on Matrix Analysis and Applications
Computing Popov and Hermite forms of polynomial matrices
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Recursiveness in matrix rational interpolation problems
Journal of Computational and Applied Mathematics - Special issue: ROLLS symposium
Shifted normal forms of polynomial matrices
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs
SIAM Journal on Matrix Analysis and Applications
On the complexity of polynomial matrix computations
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Modern Computer Algebra
Computing the rank and a small nullspace basis of a polynomial matrix
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Efficient computation of order bases
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Fraction-free row reduction of matrices of Ore polynomials
Journal of Symbolic Computation
Normal forms for general polynomial matrices
Journal of Symbolic Computation
Normalization of row reduced matrices
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]
Journal of Symbolic Computation
Efficient algorithms for order basis computation
Journal of Symbolic Computation
Computing minimal nullspace bases
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Given a matrix of univariate polynomials over a field K, its columns generate a K[x]-module. We call any basis of this module a column basis of the given matrix. Matrix gcds and matrix normal forms are examples of such module bases. In this paper we present a deterministic algorithm for the computation of a column basis of an m x n input matrix with m ≤ n. If s is the average column degree of the input matrix, this algorithm computes a column basis with a cost of Õ(nmω-1s) field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. Note that the average column degree s is bounded by the commonly used matrix degree that is also the maximum column degree of the input matrix.