Zero/pole structure of linear transfer functions

This paper studies the relationship between the zeros and poles of a linear transfer function from a module-theoretic point of view. The situation is well-understood when G(z) is proper, so that the pole module at infinity vanishes and (as always) the polynomial pole module X serves as the state space of the minimal realization (X;A,B,C) of G(z). There are isomorphisms identifying the (polynomial) zero module Z(G) with V*, the maximum (A,B)-invariant subspace of X contained in ker C, and the infinite zero module Z∞ (G) with S*, the minimum conditionally invariant subspace of X containing im B [1,2,7]. Our results here show that these two facts can be unified by using exact sequences which require no properness assumptions on G(z).