On Unimodular Matrices of Difference Operators

摘要

We consider matrices L ∈ Mat_n(K[σ,σ～(-1)]) of scalar difference operators, where K is a difference field of characteristic 0 with an automorphism σ. We discuss approaches to compute the dimension of the space of those solutions of the system of equations L(y) = 0 that belong to an adequate extension of K. On the base of one of those approaches, we propose a new algorithm for computing L～(-l) ∈ Mat_n(K[(σ,σ～(-1)]) whenever it exists. We investigate the worst-case complexity of the new algorithm, counting both arithmetic operations in K and shifts of elements of K. This complexity turns out to be smaller than in the earlier proposed algorithms for inverting matrices of difference operators. Some experiments with our implementation in Maple of the algorithm are reported.