Frontiers of Mathematics in China >

2015 , Vol. 10 >Issue 5: 1113 - 1122

DOI: https://doi.org/10.1007/s11464-015-0453-3

RESEARCH ARTICLE

An extended version of Schur-Cohn-Fujiwara theorem in stability theory

  • Yongjian HU ,
  • Xuzhou ZHAN ,
  • Gongning CHEN
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  • School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received date: 09 May 2014

Accepted date: 28 Jan 2015

Published date: 24 Jun 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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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.

Key words： Inertia of polynomial; inertia of matrix; Bezout matrix; Schur-Cohn-Fujiwara theorem; Schur-Cohn matrix

Cite this article

Yongjian HU , Xuzhou ZHAN , Gongning CHEN . An extended version of Schur-Cohn-Fujiwara theorem in stability theory[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1113 -1122 . DOI: 10.1007/s11464-015-0453-3

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