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 × 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 · 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 × 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 ¯ , by constructing a principal C ∗ × 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.