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

Abstract

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal $\mathbb {C}\times \mathbb {S}^1$-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic $\mathbb {C}$-actions $\mathscr {A}$ on the space of polynomials of degree n. For each orbit $\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}$ of $\mathscr {A}$, we study the dynamical problem of the existence of a complex rational vector field $\mathbb {X}(z)$ on $\mathbb {C}$ such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit $\{ s \cdot P(z) = 0 \}$. Regarding the above $\mathbb {C}$-action coming from the $\mathbb {C}\times \mathbb {S}^1$-bundle structure, we prove the existence of a complex rational vector field $\mathbb {X}(z)$ on $\mathbb {C}$, which describes the geometric change of the n-root configuration in the unitary disk $\mathbb {D}$ of a $\mathbb {C}$-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in $\mathbb {C}\backslash \overline{\mathbb {D}}$, by constructing a principal $\mathbb {C}^* \times \mathbb {S}^1$-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.