The number of nonunimodular roots of a reciprocal polynomial

Dragan Stankov

doi:10.5802/crmath.422

Number theory

The number of nonunimodular roots of a reciprocal polynomial

Dragan Stankov ¹

¹ Katedra Matematike RGF-a, Faculty of Mining and Geology, University of Belgrade, Belgrade, Đušina 7, Serbia

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 423-435.

Abstract
Résumé

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.

Nous introduisons une suite $P_{d}$ de polynômes réciproques unitaires à coefficients entiers ayant les coefficients centraux fixes ainsi que les coefficients périphériques. Nous prouvons que le rapport du nombre de racines non unimodulaires de $P_{d}$ sur son degré $d$ a une limite $L$ lorsque $d$ tend vers l’infini. Nous montrons que si les coefficients d’un polynôme peuvent être arbitrairement grands en module alors $L$ peut être arbitrairement proche de $0$ . Il semble raisonnable de croire que si les coefficients sont bornés, alors l’analogue de la conjecture de Lehmer est vrai : soit $L = 0$ , soit il existe un écart tel que $L$ ne puisse pas être arbitrairement proche de $0$ . Nous présentons un algorithme pour le calcul du rapport limite et une méthode numérique pour son approximation. Nous avons estimé le rapport limite pour une famille de polynômes déduits des puissances d’un nombre de Salem donné. Nous avons calculé le rapport limite des polynômes corrélés à de nombreux polynômes bivariés ayant une petite mesure de Mahler introduits par Boyd et Mossinghoff.

Received: 2022-05-03
Revised: 2022-07-19
Accepted: 2022-09-09
Published online: 2023-01-12

DOI: 10.5802/crmath.422

Author's affiliations:

Dragan Stankov ¹

¹ Katedra Matematike RGF-a, Faculty of Mining and Geology, University of Belgrade, Belgrade, Đušina 7, Serbia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Dragan Stankov},
     title = {The number of nonunimodular roots of a reciprocal polynomial},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {423--435},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.422},
     language = {en},
}

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Dragan Stankov. The number of nonunimodular roots of a reciprocal polynomial. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 423-435. doi : 10.5802/crmath.422. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.422/

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