Selected and satellite unknowns in linear differential systems

Consider the differential field K=Q‾(x) with derivation d/dx. Let some unknowns (components of the unknown vector y=(y1,…,yn)T) of a linear homogeneous differential system S over K be selected. Denote this set of selected unknowns by s. An unselected unknown yj of the system S is called satellite for s if the minimal subfield of a Picard-Vessiot extension over K for S, that contains all selected components of all solutions to S, also contains yj component of any solution. We present an algorithm for constructing the set of satellite unknowns for a given linear homogeneous differential system with selected unknowns.