A necessary and sufficient condition for an algebraic integer to be a Salem number
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 215-226.

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.

Nous donnons une condition nécessaire et suffisante pour qu’une racine strictement supérieure à 1 d’un polynôme réciproque unitaire de degré pair 4 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.

Received : 2018-05-21
Revised : 2018-06-25
Accepted : 2018-07-16
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1076
Classification:  11R06
Keywords: Salem number, -linearly independent numbers, reciprocal polynomial, Galois automorphism
@article{JTNB_2019__31_1_215_0,
     author = {Dragan Stankov},
     title = {A necessary and sufficient condition for an algebraic integer to be a Salem number},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {215-226},
     doi = {10.5802/jtnb.1076},
     language = {en},
     url={jtnb.centre-mersenne.org/item/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, Volume 31 (2019) no. 1, pp. 215-226. doi : 10.5802/jtnb.1076. https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_215_0/

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