Local realizations and local polynomial matrix representations of systems

We define the local polynomial matrix representations of a controllable matrix pair (A, B) with elements in an arbitrary field F and the local realizations of a nonsingular polynomial matrix whose elements are in F[s] with respect to a nonempty subset of Specm(F[s]). We give different characterizations of these local concepts. In particular, when F=C, local realizations and left null pairs as defined in Gohberg et al. [I. Gohberg, M.A. Kaashoek, F. van Schagen, Partially Specified Matrices and Operators: Classification, Completion, Applications, Bikhäuser, Basel, 1995] are closely related. Moreover, global polynomial matrix representations and global realizations, as defined in Zaballa [I. Zaballa, Controllability and hermite indices of matrix pairs, Int. J. Control 68 (1) (1997) 61–86] are particular cases of the same local concepts. Finally, local Wiener–Hopf factorization indices with respect to a nonempty subset of Specm(F[s]) are defined.