An algorithm to calculate the geometry of the roots of Hurwitz-symmetric polynomials

This work introduces an algorithm for analyzing the geometry of the roots of Hurwitz-symmetric polynomials, with an emphasis on counting the number of pure imaginary roots and determining their multiplicities. The approach is based on generating a Euclidean remainder sequence, which is closely related to continued fraction formulations and the Bezoutian criterion. This method provides a comprehensive framework for studying the root geometry of Hurwitz-symmetric polynomials, enabling the identification of marginal stability or instability in a broad sense. The proposed algorithm is particularly relevant in the analysis and control of linear time-invariant time-delayed systems (LTI-TDS) with commensurate delays, where the distribution of roots of the characteristic quasi-polynomial determines stability behavior. By addressing the multiplicities of pure imaginary roots, this work offers valuable insights for stability analysis and robust system design.