[Sur le nombre de racines de module un de certains polynômes réciproques]
Nous introduisons une suite de polynômes unitaires réciproques avec des coefficients entiers ayant les coefficients centraux fixes. Nous prouvons que le rapport entre le nombre de racines non unimodulaires de et son degré a une limite lorsque tend vers l’infini. Nous présentons un algorithme de calcul de la limite et une méthode numérique pour son approximation. Si est la somme d’un nombre fixe de monômes, nous déterminons les coefficients centraux de sorte que le rapport ait la limite minimale. Nous généralisons la limite du rapport pour les polynômes de plusieurs variables. Certains exemples suggèrent une conjecture pour les polynômes à deux variables qui est analogue à la formule limite de Boyd pour la mesure de Mahler.
We introduce a sequence of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of and its degree has a limit when tends to infinity. We present an algorithm for calculation the limit and a numerical method for its approximation. If is the sum of a fixed number of monomials we determine the central coefficients such that the ratio has the minimal limit. We generalise the limit of the ratio for multivariate polynomials. Some examples suggest a theorem for polynomials in two variables which is analogous to Boyd’s limit formula for Mahler measure.
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@article{CRMATH_2020__358_2_159_0,
author = {Stankov, Dragan},
title = {The number of unimodular roots of some reciprocal polynomials},
journal = {Comptes Rendus. Math\'ematique},
pages = {159--168},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {2},
year = {2020},
doi = {10.5802/crmath.28},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.28/}
}
TY - JOUR AU - Stankov, Dragan TI - The number of unimodular roots of some reciprocal polynomials JO - Comptes Rendus. Mathématique PY - 2020 SP - 159 EP - 168 VL - 358 IS - 2 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.28/ DO - 10.5802/crmath.28 LA - en ID - CRMATH_2020__358_2_159_0 ER -
%0 Journal Article %A Stankov, Dragan %T The number of unimodular roots of some reciprocal polynomials %J Comptes Rendus. Mathématique %D 2020 %P 159-168 %V 358 %N 2 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.28/ %R 10.5802/crmath.28 %G en %F CRMATH_2020__358_2_159_0
Stankov, Dragan. The number of unimodular roots of some reciprocal polynomials. Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 159-168. doi : 10.5802/crmath.28. http://www.numdam.org/articles/10.5802/crmath.28/
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