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

作　　者：Yongjian HU Xuzhou ZHAN Gongning CHEN 

机构地区：[1]School of Mathematical Sciences, Beijing Normal University Beijing 100875, China

出　　处：《Frontiers of Mathematics in China》2015年第5期1113-1122,共10页中国高等学校学术文摘·数学（英文）

基　　金：Acknowledgements This work was supported by the National Natural Science Foundation of China （Nos. 11071017, 11271045） and the Program for New Century Excellent Talents in University.

摘　　要：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.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.

关 键 词：Inertia of polynomial inertia of matrix Bezout matrix Schur-Cohn-Fujiwara theorem Schur-Cohn matrix 

分 类 号：O175.13[理学—数学] O241.1[理学—基础数学]