Efficient Computation of Order Bases

In this paper we give an efficient algorithm for computation of order basis of a matrix of power series. For a problem with an m x n input matrix over a field K, m <= n, and order sigma, Our algorithm uses O(similar to)(MM(n, [m sigma/n])) subset of O(similar to) (n(omega) [m sigma/n]) field operations in K, where the soft-O notation O(similar to) is Big-O with log factors omitted and MM (n, d) denotes the cost of multiplying two polynomial matrices with dimension n and degree d. The algorithm extends earlier work of Storjohann, whose method can be used to find a subset of an order basis that is within a specified degree bound delta using O(similar to)(MM (n, delta)) field operations for delta >= [m sigma/n].