  Multidimensional systems are becoming increasingly important as they provide
a promising tool for estimation, simulation and control, while going beyond the
traditional setting of one-dimensional systems. The analysis of
multidimensional systems is linked to multivariate polynomials, and is
therefore more difficult than the well-known analysis of one-dimensional
systems, which is linked to univariate polynomials. In the current paper we
relate the realization theory for overdetermined autonomous multidimensional
systems to the problem of solving a system of polynomial equations. We show
that basic notions of linear algebra suffice to analyze and solve the problem.
The difference equations are associated with a Macaulay matrix formulation, and
it is shown that the null space of the Macaulay matrix is a multidimensional
observability matrix. Application of the classical shift trick from realization
theory allows for the computation of the corresponding system matrices in a
multidimensional state-space setting. This reduces the task of solving a system
of polynomial equations to computing an eigenvalue decomposition. We study the
occurrence of multiple solutions, as well as the existence and analysis of
solutions at infinity, which allow for an interpretation in terms of
multidimensional descriptor systems.
