Proper, Minimal Macmillan Degree, Bases of Rational Vector Spaces

The algebraic structure of the set of all proper rational vectors contained in a given rational vector space ¿(s) is shown to be that of a Noetherian Rpr(s)-module M* (Rpr(s) = the ring of proper rational functions). The proper submodules Mi of M* form an ascending chain of submodules partially ordered by an invariant of Mi defined as the valuation at s = ¿ of Mi. The various bases of Mi are examined and classified according to their property of column reduceness at s = ¿. The concept of a prime column reduced at s = ¿ basis of Mi is introduced. It is shown that prime bases of Mi can be further classified by their MacMillan degrees and the existence of minimal MacMillan degree bases for Mi is established. A prime and minimal MacMillan degree basis of Mi extends Forney\´s concept of a minimal polynomial basis of ¿(s) for the Rpr(s)-module Mi. The MacMillan degrees of the columns of such bases form a set of invariants for Mi which are defined as the "generalized invariant dynamical indices" of Mi, and a simple relation is established between (i) the generalized invariant dynamical indices" of Mi, (ii) the orders of zeros at s = ¿ of Mi, and (iii) the Forney invariant dynamical indices of ¿(s).