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.

翻译：给定单个代数输入-输出方程，我们提出了一种方法，以有理实现的形式寻找关联系统的不同表示方法；这些是右侧为有理函数的动力系统。已有研究表明，当输入-输出方程为一阶时，若存在有理实现则可被计算得出。本文首先关注所谓可观测有理实现的存在性与实际计算，其次研究具有实系数的有理实现。对可观测实现的研究使我们能够找到给定一阶输入-输出方程的所有有理实现，以及此过程中必要的域扩张。我们证明，对于一阶输入-输出方程，有理实现的存在性等价于可观测有理实现的存在性。此外，我们给出了判定实有理实现存在性的准则。一阶输入-输出方程的可观测实现与实实现的计算是完全算法化的。我们还给出了高阶输入-输出方程情况的初步结果。