Salem numbers as Mahler measures of nonreciprocal units

Artūras Dubickas¹

¹ Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania

Acta Arithmetica, Tome 176 (2016) no. 1, pp. 81-88

Résumé

We show that for $d=4$ and for each $d=4\ell+2$, where $\ell \in \mathbb N$, there are Salem numbers of degree $d$ which belong to the set of nonreciprocal Mahler measures $L_0$. In passing, we show that for every odd $n$ there exist Salem polynomials $f$ of degree $d=2n$ whose Galois group is isomorphic to $\mathbb Z_2^{n-1}\rtimes G_g$, where $G_g$ is the Galois group of the trace polynomial $g$ of $f$. The first result addresses a corresponding question of Boyd, whereas the second result is related to and in some sense completes an earlier result of Christopoulos and McKee.

Détail

DOI : 10.4064/aa8407-8-2016

Keywords: each ell where ell mathbb there salem numbers degree which belong set nonreciprocal mahler measures passing every odd there exist salem polynomials degree whose galois group isomorphic mathbb n rtimes where galois group trace polynomial nbsp first result addresses corresponding question boyd whereas second result related sense completes earlier result christopoulos mckee