Zero location by Hermitian forms: The singular case

Abstract

The Schur-Cohn criterion for the number of zeros of a polynomial inside and outside the unit disc fails if the polynomial has a pair of conjugate zeros or a zero on the unit circle: the corresponding quadratic form is singular. Recently the authors have shown [5] that the classical Schur-Cohn criterion may be deduced from a simple algebraic identity; this yields not only a very simple proof but also a substantial generalization. The method produces a whole family of quadratic forms which may be used for testing the zeros. In the present paper the same algebraic identity is used to show that singularity of these quadratic forms is always due to the presence of pairs of conjugate zeros or zeros on the unit circle. There is a method for ascertaining zero distribution in the singular case by differentiation; we give a derivation of this test on the basis of our matrix-theoritic treatment. The second section deals with the same problem in the case of a general circle or half plane.