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.

我们为一个复杂变量的Schur稳定多项式提供了Schur-Cohn稳定性算法的详细研究。首先，实解析主要\（\ mathbb {C} \倍\ mathbb {S} ^ 1 \）在家庭程度的舒尔稳定多项式-bundle结构Ñ构造。其次，我们考虑阶数为n的多项式空间上的全纯\（\ mathbb {C} \）-作用\（\ mathscr {A} \）。对于每一个轨道\（\ {S \ CDOT P（z）的\ \ VERT \ S \在\ mathbb {C} \} \）的\（\ mathscr {A} \） ，我们研究的存在的动力学问题\（\ mathbb {C} \）上的复数有理向量场\（\ mathbb {X}（z）\）这样，其全纯s时间描述了轨道\（\ {s \ cdot P（z）= 0 \} \）的n根构型的几何变化。关于来自\（\ mathbb {C} \ times \ mathbb {S} ^ 1 \） -bundle结构的上述\（\ mathbb {C} \）作用，我们证明了复有理矢量场\的存在。（\ mathbb {X}（Z）\）上\（\ mathbb {C} \） ，它描述了的几何变化ñ -root配置在单一的磁盘\（\ mathbb {d} \）的\（ \ mathbb {C} \）-Schur稳定多项式的轨道。我们在反Schur多项式的框架中获得了并行结果，这些结果的所有根都来自\（\ mathbb {C} \反斜杠\ overline {\ mathbb {D}} \），通过构造一个主体\（\ mathbb {C } ^ * \ times \ mathbb {S} ^ 1 \） -此多项式族中的束结构。作为队列人口模型的应用，描述了舒尔稳定性的研究和舒尔稳定性丧失的标准。