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/ |
| Summary: | 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. |
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| ISSN: | 1778-3569 |