On numbers which are Mahler measures
Let \(\alpha\) be an algebraic number of degree \(d \geq 2\) with minimal polynomial \(P(z) = a_0 z^d + \dots + a_d \in {\mathbb Z}[z]\) over the rationals, and with conjugates \(\alpha_1, \dots, \alpha_d\). Then the Mahler measure of \(\alpha\) is defined by \(M(\alpha) = | a_0| \prod_{j=1}^d \max(1,| \alpha_j| )\). It is well known that \(\beta = M(\alpha)\) is an algebraic integer. One calls such \(\beta\) \textit{Mahler measures}. In this paper, the author continues the study of the question of which algebraic integers \(\beta\) are Mahler measures. \textit{R. L. Adler} and \textit{B. Marcus} [Mem. Am. Math. Soc. 219 (1979; Zbl 0412.54050)] showed that a Mahler measure must be a Perron number, a positive real algebraic integer that is strictly larger than the absolute value of its Galois conjugates, but the reviewer [Ergodic Theory Dyn. Syst. 6, 485--488 (1986; Zbl 0591.12003)], showed that not all Perron numbers are Mahler measures. The author extends the criteria in the latter paper to prove a number of interesting results. One of these results is that for every Perron number \(\beta\), there is some integer multiple \(n \beta\) that is a Mahler measure. He proves that if \(\beta\) is a Mahler measure then so is \(\beta^n\) for every \(n \geq 1\) (note that this is \textit{not} obvious since if \(\beta = M(\alpha)\), then in case the degree of \(\alpha^n\) is less than that of \(\alpha\), it is not true that \(\beta^n = M(\alpha^n)\)). The criteria established are not useful for reciprocal \(\beta\) (those for which \(1/\beta\) is a conjugate of \(\beta\)). By special arguments, the author establishes the following interesting result: if \(\beta\) and \(\gamma\) are reciprocal quadratic units with \(\gamma \notin {\mathbb Q}(\beta)\) then \(\beta\gamma\) is not the Mahler measure of a reciprocal algebraic number.
scientific article; zbMATH DE number 2102087
scientific article