On real and observable realizations of input-output equations

Given a single algebraic input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand sides. It has been shown that in the case where the input-output equation is of order one, rational realizations can be computed, if they exist. In this work, we focus first on the existence and actual computation of the so-called observable rational realizations, and secondly on rational realizations with real coefficients. The study of observable realizations allows to find every rational realization of a given first order input-output equation, and the necessary field extensions in this process. We show that for first order input-output equations the existence of a rational realization is equivalent to the existence of an observable rational realization. Moreover, we give a criterion to decide the existence of real rational realizations. The computation of observable and real realizations of first order input-output equations is fully algorithmic. We also present partial results for the case of higher order input-output equations.