Coprime factorizations of multivariate rational matrices

Coprime factorization is a well-known issue in one-dimensional systems theory, having many applications in realization theory, balancing, controller synthesis, etc. Generalization to systems in more than one independent variable is a delicate matter: First, several nonequivalent coprimeness notions for multivariate polynomial matrices have been discussed in the literature: zero, minor, and factor coprimeness. Here we adopt a generalized version of factor primeness that appears to be most suitable for multidimensional systems: a matrix is prime iff it is a minimal annihilator. After reformulating the sheer concept of a factorization, it is shown that every rational matrix possesses left and right coprime factorizations that can be found by means of computer algebraic methods. Several properties of coprime factorizations are given in terms of certain determinantal ideals.