On the root distribution of general polynomials with respect to the unit circle

Abstract

A modification and an extension of Schur-Cohn's algorithm and Jury's table are presented. The new versions of the classical algorithms allow us to associate a sequence of coefficients k j to every polynomial P. We establish that the number of zeros of P outside the unit circle equals the number of k j satisfying ¦k j¦ > 1 . A simple expression for the number of zeros of P on the unit circle is also established. An extension of the modified algorithm which introduces an arbitrary parameter allows us to study the critical situations: 1 − ε ⩽ ¦k j¦ ⩽ 1 + ε , where ε is an arbitrary small positive number.