Abstract
This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.
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Hu, Y., Zhan, X. & Chen, G. An extended version of Schur-Cohn-Fujiwara theorem in stability theory. Front. Math. China 10, 1113–1122 (2015). https://doi.org/10.1007/s11464-015-0453-3
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DOI: https://doi.org/10.1007/s11464-015-0453-3