Nous donnons une condition nécessaire et suffisante pour qu’une racine strictement supérieure à d’un polynôme réciproque unitaire de degré pair à coefficients entiers soit un nombre de Salem. Cette condition exige que le polynôme minimal d’une certaine puissance de cet entier algébrique ait un coefficient linéaire assez grand. Pour les nombres de Salem de certains petits degrés nous déterminons également la probabilité qu’une puissance d’un tel nombre satisfasse à cette condition.
We present a necessary and sufficient condition for a root greater than unity of a monic reciprocal polynomial of an even degree at least four, with integer coefficients, to be a Salem number. This condition requires that the minimal polynomial of some power of the algebraic integer has a linear coefficient that is relatively large. We also determine the probability that an arbitrary power of a Salem number, of certain small degrees, satisfies this condition.
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DOI : 10.5802/jtnb.1076
Mots clés : Salem number, $\protect \mathbb{Q}$ -linearly independent numbers, reciprocal polynomial, Galois automorphism
@article{JTNB_2019__31_1_215_0,
author = {Stankov, Dragan},
title = {A necessary and sufficient condition for an algebraic integer to be a {Salem} number},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {215--226},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {31},
number = {1},
year = {2019},
doi = {10.5802/jtnb.1076},
mrnumber = {3994727},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.1076/}
}
TY - JOUR AU - Stankov, Dragan TI - A necessary and sufficient condition for an algebraic integer to be a Salem number JO - Journal de théorie des nombres de Bordeaux PY - 2019 SP - 215 EP - 226 VL - 31 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.1076/ DO - 10.5802/jtnb.1076 LA - en ID - JTNB_2019__31_1_215_0 ER -
%0 Journal Article %A Stankov, Dragan %T A necessary and sufficient condition for an algebraic integer to be a Salem number %J Journal de théorie des nombres de Bordeaux %D 2019 %P 215-226 %V 31 %N 1 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.1076/ %R 10.5802/jtnb.1076 %G en %F JTNB_2019__31_1_215_0
Stankov, Dragan. A necessary and sufficient condition for an algebraic integer to be a Salem number. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 215-226. doi : 10.5802/jtnb.1076. http://www.numdam.org/articles/10.5802/jtnb.1076/
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