Efficient algorithms for order basis computation

In this paper we present two algorithms for the computation of a shifted order basis of an m×n matrix of power series over a field K with m ≤ n. For a given order σ and balanced shift ~s the first algorithm determines an order basis with a cost of O∼(nωdmσ/ne) field operations in K, where ω is the exponent of matrix multiplication. Here an input shift is balanced when max(~s)−min(~s) ∈ O(mσ/n). This extends earlier work of Storjohann which only determines a subset of an order basis that is within a specified degree bound δ using O∼(nωδ) field operations for δ ≥ dmσ/ne. While the first algorithm addresses the case when the column degrees of a complete order basis are unbalanced given a balanced input shift, it is not efficient in the case when an unbalanced shift results in the row degrees also becoming unbalanced. We present a second algorithm which balances the high degree rows and computes an order basis also using O∼(nωdmσ/ne) field operations in the case that the shift is unbalanced but satisfies the condition ∑n i=1(max(~s)−~si) ≤ mσ. This condition essentially allows us to locate those high degree rows that need to be balanced. This extends the earlier work by the authors from ISSAC’09.