Constructing matrices with given generalized characteristic polynomial

Generalized characteristic polynomials, a standard tool in 2D control theory, are treated from a symbolic point of view. The main cases of the multidimensional realization problem can be handled, even with parameters rather than explicit numbers, up to dimension at least 5 with an implementation of Janet's algorithm. For the bivariate case a general existence result is presented. The problem to parametrize all realizations can be split up into a sequence of smaller problems, each of which can be tackled with the software available. We concentrate on the generic cases. Various coefficients in the generalized characteristic polynomial are interpreted as (generalized) characteristic polynomials of submatrices, and their expansion in multivariate Lagrange interpolation bases turns out to be relevant.