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објављена верзија

енглески

Dragan Stankov

The number of nonunimodular roots of a reciprocal polynomial

COMPTES RENDUS MATHEMATIQUE

Elsevier France ^Editions Scientifiques et Medicales

2023

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 of the number of
nonunimodular roots of $P_{d}$ to 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 calculating the limit ratio and a numerical method for its approximation.
We estimated the limit ratio for a family of polynomials deduced from the powers of a given Salem number. We calculated the limit ratio of polynomials correlated to many bivariate polynomials having small Mahler measure introduced by Boyd and Mossinghoff.

Volume 361 (2023)

361 (G1)

423

435

10.5802/crmath.422

1631-073X e-ISSN:1778-3569

Algebraic integer; the house of algebraic integer; maximal modulus; reciprocal polynomial; primitive polynomial, Schinzel-Zassenhaus conjecture; Mahler measure; method of least squares; cyclotomic polynomials

Algebraic integer; the house of algebraic integer; maximal modulus; reciprocal polynomial; primitive polynomial, Schinzel-Zassenhaus conjecture; Mahler measure; method of least squares; cyclotomic polynomials

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.5802/crmath.422.pdf

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