Computing column bases of polynomial matrices

Abstract

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(nm^ω-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.