On numbers which are Mahler measures
type: Item
instance of: scholarly article
title: On numbers which are Mahler measures
zbMATH Open document ID: 1065.11084
published in: Monatshefte für Mathematik
publication date: 2004-09-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 2102087
zbMATH Keywords: Mahler measure
zbMATH Keywords: Perron number
author: A. Dubickas
reviewed by: David W. Boyd
MaRDI profile type: Publication
Recommended article: Mahler measures of Pisot and Salem type numbers
Recommended article: On values of the Mahler measure in a quadratic field (solution of a problem of Dixon and Dubickas)
Recommended article: Every Real Algebraic Integer Is a Difference of Two Mahler Measures
Recommended article: Reciprocal Algebraic Integers Whose Mahler Measures are Non-Reciprocal
Recommended article: Mahler Measures Close to an Integer
Recommended article: Mahler measures generate the largest possible groups
Recommended article: Mahler measures in a cubic field
Recommended article: Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers
Recommended article: The Mahler measure and its areal analog for totally positive algebraic integers
Recommended article: Mahler measure of a difference of two conjugates
description: scientific article; zbMATH DE number 2102087