Stability Region for Discrete Time Systems and Its Boundary

In this paper we investigate the Schur stability region of the $n$th order polynomials in the coefficient space. Parametric description of the boundary set is obtained. We show that all the boundary can be obtained as a multilinear image of three $(n-1)$-dimensional boxes. For even and odd $n$ these boundary boxes are different. Analogous properties for the classical multilinear reflection map are unknown. It is shown that for $n \geq 4$, both two parts of the boundary which are pieces of the corresponding hyperplanes are nonconvex. Polytopes in the nonconvex stability region are constructed. A number of examples are provided.