Linear relations with conjugates of a Salem number
Artūras Dubickas1 ; Jonas Jankauskas2
1 Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University Naugarduko 24 03225 Vilnius, Lithuania
2 Mathematik und Statistik Montanuniversität Leoben Franz Josef Strasse 18 8700 Leoben, Austria
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 179-191.

Dans cet article, nous considérons les relations linéaires entre les conjugués d’un nombre de Salem α. Nous montrons qu’une telle relation provient d’une relation linéaire entre les conjugués de l’entier algébrique totalement réel correspondant α+1/α. On montre également que le plus petit degré d’un nombre de Salem satisfaisant à une relation non triviale entre ces conjugués est 8 tandis que la longueur la plus courte d’une relation linéaire non-triviale entre les conjugués d’un nombre de Salem est 6.

In this paper we consider linear relations with conjugates of a Salem number α. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer α+1/α. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is 8, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is 6.

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DOI : 10.5802/jtnb.1116
Classification : 11R06, 11R09
Keywords: linear additive relations, Salem numbers, Pisot numbers, totally real algebraic numbers
Mots clés : Les relations linéaires additives, les nombres de Salem, les nombres de Pisot, les nombres algébriques totalement réels
Artūras Dubickas 1 ; Jonas Jankauskas 2

1 Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University Naugarduko 24 03225 Vilnius, Lithuania
2 Mathematik und Statistik Montanuniversität Leoben Franz Josef Strasse 18 8700 Leoben, Austria
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Artūras Dubickas; Jonas Jankauskas. Linear relations with conjugates of a Salem number. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 179-191. doi : 10.5802/jtnb.1116. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1116/

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