Cascade representation and minimal factorization of multidimensional systems

We give necessary and sufficient conditions for both the existence of minimal factorization of a rational function of NN complex variables and the cascade representation of a Fornasini–Marchesini linear system. In particular, we generalize the well known result of Bart, Gohberg, Kaashoek and Van Doren on minimal factorization of a rational function of one complex variable.