On the Roots of a Family of Polynomials

Abstract

The aim of this paper is to give a characterization of the set of roots of a special family of polynomials. This family is relevant in reliability theory since it contains the reliability polynomials of the networks created by series-parallel compositions. We prove that the set of roots is bounded, being contained in the two disks of the radius equal to the golden ratio, centered at 0 and at 1. We study the closure of the set of roots and prove that it includes two disks centered at 0 and 1 of a radius slightly greater than 1, as well as the sinusoidal spirals centered at 0 and at 1, respectively. The expression of some limit points is also provided.

The Roots of the Reliability Polynomials of Circular Consecutive-k-out-of-n:F Systems

“…A more general result [25] states that if all the (complex) roots of a polynomial with positive coefficients lie in the region 2π 3 < arg z < 4π 3 , then the sequence of coefficients is log-concave. Finding the set of roots of various polynomials related to different families of graphs is both a tempting and also a fruitful problem which has attracted lots of research in the past few decades (see, for instance [22,[26][27][28][29][30] for reliability polynomials; [31][32][33][34][35] for chromatic polynomials; and [36][37][38][39] for independence polynomials). In many of these papers, a central role is played by a theorem on analytic functions due to Beraha, Kahane, and Weiss [40], which proves to be a highly useful instrument to study the zeros of a family of polynomials defined recursively.…”