Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers

Zaïmi, Toufik

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 877-889.

Motivé par une question de M. J. Bertin, on obtient des paramétrisations des polynômes minimaux des nombres de Salem quartiques, disons $\alpha$, qui sont des mesures de Mahler des $2$ -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres $\alpha$, de trace donnée, et de déduire que pour tout entier naturel $t$ (resp. $t\ge 2)$, il y a un nombre de Salem quartique, de trace $t$, qui est (resp. qui n’est pas) une mesure de Mahler d’un $2$ -nombre de Pisot non-réciproque.

Motivated by a question of M. J. Bertin, we obtain parametrizations of minimal polynomials of quartic Salem numbers, say $\alpha$, which are Mahler measures of non-reciprocal $2$-Pisot numbers. This allows us to determine all such numbers $\alpha$ with a given trace, and to deduce that for any natural number $t$ (resp. $t\ge 2)$ there is a quartic Salem number of trace $t$ which is (resp. which is not) a Mahler measure of a non-reciprocal $2$-Pisot number.

Classification : 11R06, 11R80, 11J71
Mots clés : Salem numbers, Mahler measure, $2$-Pisot numbers.

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