## 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 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.

Original language | English |
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Pages (from-to) | 545-587 |

Number of pages | 43 |

Journal | Mathematics of Control, Signals, and Systems |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2019 |

### Bibliographical note

Publisher Copyright:© 2019, Springer-Verlag London Ltd., part of Springer Nature.

## Keywords

- Complex rational vector fields
- Lie group actions
- Principal G-bundles
- Schur stable polynomials
- Schur–Cohn stability algorithm