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.