Note on roots location of a symmetric polynomial with respect to the imaginary axis

Abstract

In the theory of the separation of roots of algebraic equations, the well-known Routh–Hurwitz–Fujiwara theorem enables us to separate the complex roots of a polynomial with complex coefficients in terms of the inertia of a related Hermitian matrix. Unfortunately, it fails if the polynomial has a nontrivial factor which is symmetric with respect to the imaginary axis. In this article, we present a method to overcome the fault and formulate the inertia of a scalar polynomial with complex coefficients in terms of the inertia of several Hermitian matrices based on a factorization of a monic symmetric polynomial into products of monic symmetric polynomials with only simple roots in the complex plane and on computing the inertia of each factor by means of a subtle perturbation.

Keywords:

• inertia of a matrix
• inertia of a polynomial
• Routh–Hurwitz problem
• Bezoutian matrix
• Routh–Hurwitz–Fujiwara theorem

AMS Subject Classifications::

• 12D10
• 15A57

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11071017) and the Program for New Century Excellent Talents in University. The authors are very grateful to the anonymous reviewer and editors for their valuable comments and suggestions for improving the quality of this article.