The number of nonunimodular roots of a reciprocal polynomial

TL;DR

This paper studies the asymptotic behavior of nonunimodular roots of a specific class of reciprocal polynomials, providing theoretical results, an algorithm for limit calculation, and numerical estimates related to Mahler measure.

Contribution

It introduces a sequence of reciprocal polynomials, proves the existence of a limit ratio of nonunimodular roots, and develops methods to compute and estimate this limit.

Findings

The ratio of nonunimodular roots to degree tends to a limit L as degree increases.

If polynomial coefficients are unbounded, L can be arbitrarily close to 0.

Numerical estimates of L are provided for various polynomial families.

Abstract

We introduce a sequence P_d of monic reciprocal polynomials with integer coefficients having the central coefficients fixed as well as the peripheral coefficients. We prove that the ratio between number of nonunimodular roots of P_d and its degree d has a limit L when d tends to infinity. We show that if the coefficients of a polynomial can be arbitrarily large in modulus then L can be arbitrarily close to 0. It seems reasonable to believe that if the coefficients are bounded then the analogue of Lehmer's Conjecture is true: either L = 0 or there exists a gap so that L could not be arbitrarily close to 0. We present an algorithm for calculation the limit ratio and a numerical method for its approximation. We estimated the limit ratio for a family of polynomials. We calculated the limit ratio of polynomials correlated to many bivariate polynomials having small Mahler measure.