Notes on computing minimal approximant bases

When s = 1 and N = ‖n‖ − 1 this is the classical Hermite Padé approximation problem. Here we allow N to be arbitrary. We describe algorithms for computing an order N genset of type n: a matrix V ∈ k[x]∗×m such that every row of V is a solution to (1) and every solution P of (1) can be expressed as a k[x]-linear combination of the rows of V . Ideally, V will be a minbasis of solutions: V has full row rank, and if n̄ ≥ maxi ni then V diag(n̄−n1, . . . , n̄−nm) is row reduced (e.g., in weak Popov form). To compare with [1], an order N minbasis of type n will be comprised of those rows of a σ-basis (with σ = sN) which satisfy the degree constraints (i.e., have positive defect), and vice versa. For example, the Popov form of the