Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].