On Solution Spaces of Products of Linear Differential or Difference Operators
Résumé
We consider linear ordinary differential or difference systems of the form L(y) = 0 where L is an operator with matrix coefficients, the unknown vector y has m components y1, . . . , ym, m > 1. The matrix coefficients are of size m x m, their entries belong to a differential or difference field K of characteristic 0. For any such a system the solution space VL is considered, and the components of each solution are in a fixed appropriate differential or difference extension of K (e.g., in the universal Picard-Vessiot extension). We prove that dim VLM = dim VL+dim VM for arbitrary operators L and M of the considered form, and discuss some algorithms based on this property of operators. In particular, we propose an algorithm to compute dim VL, as well as a new algorithm having a low complexity for recognizing unimodular operators and constructing the inverse of a unimodular operator.