Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal C× S¹-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic C-actions A on the space of polynomials of degree n. For each orbit {s·P(z)|s∈C} of A, we study the dynamical problem of the existence of a complex rational vector field X(z) on C such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit { s· P(z) = 0 }. Regarding the above C-action coming from the C× S¹-bundle structure, we prove the existence of a complex rational vector field X(z) on C, which describes the geometric change of the n-root configuration in the unitary disk D of a C-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in C\ D¯ , by constructing a principal C^∗× S¹-bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.