Coprimeness in the ring of psedorational transfer functions

The class of pseudorational transfer functions, roughly speaking, consists of the ratio of entire functions of exponential type that are Laplace transforms of distributions with compact support. This class is known to include all delay-differential systems, and other interesting distributed parameter systems. Characterizing a pair that satisfies the Bezout identity in this ring is an important question in various aspects: finding minimal realizations, direct sum decompositions for behaviors, stabilizing compensator parametrization, etc. Existence of a Bezout condition in this ring is deeply connected with charactering the maximal ideal space of this ring. It is also closely related to the Gelfand representation theorem for a Banach algebra. We give a characterization of the maximal ideal space of this ring (at least for a generic case of simple zeros only), and also prove that a pair (p, q) satisfies a Bezout condition if the corresponding Laplace transforms have no (finite) common zeros and also possess no "cancellation at infinity".