Testing reachability and stabilizability of systems over polynomial rings using Gröbner bases

Conditions for the reachability and stabilizability of systems over polynomial rings are well-known in the literature. For a system $ \Sigma = (A,B)$ they can be expressed as right-invertibility cconditions on the matrix $(zI - A \mid B)$. Therefore there is quite a strong algebraic relationship between both conditions, but unfortunately they are difficult to check explicitly. In this paper we introduce for each system $ \Sigma = (A,B)$ a corresponding polynomial ideal I which characterizes both reachability and stabilizability in a very straightforward way. Moreover, methods are given to compute this ideal and its variety explicitly using Gröbner Bases techniques. With help of the Gröbner Basis of the ideal I, conclusions on the reachability and stabilizability of the system $ \Sigma = (A,B)$ are easy to draw.