Approximation of the number of nonunimodular zeros of a self-reciprocal polynomial

Abstract

We introduce the ratio of the number of roots, not equal to 1 in modulus, of a reciprocal polynomial $R_d(x)$ to its degree $d$. For some sequences of reciprocal polynomials we show that the ratio has a limit $L$ when $d$ tends to infinity. Each of these sequences is defined using a two-variable polynomial $P(x,y)$ so that $R_d(x) = P(x,x^n)$. We present a few methods for approximation of the limit ratio, some of them we originally developed. In a previous paper we have calculated the exact value of the limit ratio of polynomials, correlated to many bivariate polynomials, of degree two in $y$, having small Mahler measure listed by Boyd and Mossinghoff. Now we approximate the limit ratio of $P(y^n,y)$, for all polynomials in the list. Knowing some exact values of the limit ratio, we compare the error of the approximation and the time duration of these methods. We show that the limit ratio of the sequence $P(x,x^n)$ usually is not equal to the limit ratio of the sequence $P(y^n,y)$, unlike the Mahler measure.