The number of nonunimodular roots of a reciprocal polynomial
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...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.422/ |
| _version_ | 1797651484198830080 |
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| author | Stankov, Dragan |
| author_facet | Stankov, Dragan |
| author_sort | Stankov, Dragan |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T16:16:27Z |
| format | Article |
| id | doaj.art-989e08e240794cfd908ed58cc1336c86 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:27Z |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-989e08e240794cfd908ed58cc1336c862023-10-24T14:20:20ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G142343510.5802/crmath.42210.5802/crmath.422The number of nonunimodular roots of a reciprocal polynomialStankov, Dragan0Katedra Matematike RGF-a, Faculty of Mining and Geology, University of Belgrade, Belgrade, Đušina 7, SerbiaWe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.422/ |
| spellingShingle | Stankov, Dragan The number of nonunimodular roots of a reciprocal polynomial Comptes Rendus. Mathématique |
| title | The number of nonunimodular roots of a reciprocal polynomial |
| title_full | The number of nonunimodular roots of a reciprocal polynomial |
| title_fullStr | The number of nonunimodular roots of a reciprocal polynomial |
| title_full_unstemmed | The number of nonunimodular roots of a reciprocal polynomial |
| title_short | The number of nonunimodular roots of a reciprocal polynomial |
| title_sort | number of nonunimodular roots of a reciprocal polynomial |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.422/ |
| work_keys_str_mv | AT stankovdragan thenumberofnonunimodularrootsofareciprocalpolynomial AT stankovdragan numberofnonunimodularrootsofareciprocalpolynomial |