We derive an algorithm that computes the state difference equations for a given set of poles of linear discrete overdetermined autonomous mD systems. These difference equations allow the realization of the dynamical system by means of delay, multiplication and addition elements in simulation diagrams. In doing so we generalize the classical Cayley-Hamilton theorem to multivariate polynomial ideals and provide a system theoretic interpretation to the notion of polynomial ideals, leading monomials and Grobner bases. Furthermore, we extend the problem to include poles at infinity and so arrive at a description of overdetermined descriptor systems. This results in a new state space description of autonomous mD descriptor systems. In addition, we discuss the separation of the state variables of singular mD systems into a regular and singular part. A sufficient condition under which these two state vector parts can be interpreted as a forward evolving regular part and a backward evolving singular part is given. The robustness and efficiency of the developed algorithms are demonstrated via numerical experiments. (C) 2015 Elsevier Ltd. All rights reserved.