The number of nonunimodular roots of a reciprocal polynomial

Dragan Stankov

doi:10.5802/crmath.422

Théorie des nombres

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.

Résumé
Abstract

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.

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.

Reçu le : 2022-05-03
Révisé le : 2022-07-19
Accepté le : 2022-09-09
Publié le : 2023-01-12

DOI : 10.5802/crmath.422

Affiliations des auteurs :

Dragan Stankov ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G1_423_0,
     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},
}

TY  - JOUR
AU  - Dragan Stankov
TI  - The number of nonunimodular roots of a reciprocal polynomial
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 423
EP  - 435
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.422
LA  - en
ID  - CRMATH_2023__361_G1_423_0
ER  -

%0 Journal Article
%A Dragan Stankov
%T The number of nonunimodular roots of a reciprocal polynomial
%J Comptes Rendus. Mathématique
%D 2023
%P 423-435
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.422
%G en
%F CRMATH_2023__361_G1_423_0

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/

Bibliographie
Cité par

[1] Peter Borwein; Stephen Choi; Ron Ferguson; Jonas Jankauskas On Littlewood polynomials with prescribed number of zeros inside the unit disk, Can. J. Math., Volume 67 (2015) no. 3, pp. 507-526 | DOI | MR | Zbl

[2] Peter Borwein; Tamás Erdélyi; Ron Ferguson; Richard Lockhart On the zeros of cosine polynomials: solution to a problem of Littlewood, Ann. Math., Volume 167 (2008) no. 3, pp. 1109-1117 | DOI | MR | Zbl

[3] David W. Boyd; Michael J. Mossinghoff Small limit points of Mahler’s measure, Exp. Math., Volume 14 (2005) no. 4, pp. 403-414 | DOI | MR | Zbl

[4] Paulius Drungilas Unimodular roots of reciprocal Littlewood polynomials, J. Korean Math. Soc., Volume 45 (2008) no. 3, pp. 835-840 | DOI | MR | Zbl

[5] Pál Erdős; Pál Turán On the Distribution of Roots of Polynomials, Ann. Math., Volume 51 (1950) no. 1, pp. 105-119 | DOI | MR | Zbl

[6] Valérie Flammang The N-measure for algebraic integers having all their conjugates in a sector, Rocky Mt. J. Math., Volume 50 (2020) no. 6, pp. 2035-2045 | MR | Zbl

[7] Valérie Flammang The S-measure for algebraic integers having all their conjugates in a sector, Rocky Mt. J. Math., Volume 50 (2020) no. 4, pp. 1313-1321 | MR | Zbl

[8] Christelle Guichard; Jean-Louis Verger-Gaugry On Salem numbers, expansive polynomials and Stieltjes continued fractions, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 3, pp. 769-804 | DOI | Numdam | MR | Zbl

[9] James McKee; Chris Smyth Around the Unit Circle. Mahler measure, integer matrices and roots of unity., Universitext, Springer, 2021 | DOI | Zbl

[10] Keshav Mukunda Littlewood Pisot numbers, J. Number Theory, Volume 117 (2006) no. 1, pp. 106-121 | DOI | MR | Zbl

[11] Raphael Salem Power series with integral coefficients, Duke Math. J., Volume 12 (1945), pp. 153-172 | MR | Zbl

[12] Raphael Salem Algebraic numbers and Fourier analysis, D. C. Heath and Company, 1963 | MR | Zbl

[13] Dragan Stankov The number of unimodular roots of some reciprocal polynomials, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 2, pp. 159-168 | Numdam | MR | Zbl

Cité par Sources :

Commentaires - Politique