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 x S(1-)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 center dot P(z) | s is an element of 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 center dot P(z)=0}. Regarding the above C-action coming from the C x S(1-)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) over bar, by constructing a principal C* x S(1-)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.