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arXiv · 2203.07710

The number of nonunimodular roots of a reciprocal polynomial

Dragan Stankov
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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.

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Dragan Stankov. 2022-03-15. The number of nonunimodular roots of a reciprocal polynomial. https://arxiv.org/abs/2203.07710

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