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.
. [J]. Frontiers of Mathematics in China, 2015, 10(5): 1113-1122.
Yongjian HU,Xuzhou ZHAN,Gongning CHEN. An extended version of Schur-Cohn-Fujiwara theorem in stability theory. Front. Math. China, 2015, 10(5): 1113-1122.
Fujiwara M. über die Wurzelanzahl algebraischer Gleichungen innerhalb und auf dem Einheitskreis. Math Z, 1924, 19: 161-169
6
Heinig G, Rost K. Algebraic Methods for Toeplitz-like Matrices and Operators. Operator Theory, Vol 13, Basel: Birkh?user, 1984
https://doi.org/10.1007/978-3-0348-6241-7
7
Holtz O, Tyaglov M. Structured matrices, continued fractions, and root localization of polynomial. SIAM Review, 2012, 54: 421-509
https://doi.org/10.1137/090781127
8
Krein M G. To the theory of symmetric polynomials. Mat Sb, 1933, 40(3): 271-283
9
Krein M G, Naimark M A. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear Multilinear Algebra, 1981, 10: 265-308
https://doi.org/10.1080/03081088108817420
10
Lancaster P, Tismenetsky M. The Theory of Matrices with Applications. 2nd ed. New York: Academic Press, 1985
