The Lehmer constants of an annulus
type: Item
instance of: scholarly article
title: The Lehmer constants of an annulus
zbMATH Open document ID: 1030.11057
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2002-07-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93819
full work available at URL: jourelec
review text: The Lehmer constant \(L(V)\) of an open set \(V\) in the complex plane is defined [\textit{M. Langevin}, C. R. Acad. Sci., Paris, Sér I. 301, 463-466 (1985; Zbl 0585.12013)] by \(L(V)=\inf M(\alpha)^{1/\deg(\alpha)}\), the infimum taken over all non-zero non-cyclotomic algebraic integers \(\alpha\) lying with all conjugates outside \(V\), and \(M(\alpha)\) denotes the Mahler measure of \(\alpha\). The authors compute \(L(V)\) in the case when \(V\) is an annulus centered at the origin and prove that for any open set \(V\) the inequality \(L(V)>1\) implies that \(V\) has a non-empty intersection with the unit circle (the converse implication has been earlier established by Langevin (\textit{loc.cit}.)) Moreover they consider a variant of Lehmer's constant, namely \(L_\infty(V)=\lim_{d\to\infty}\inf M(\alpha)^{1/\deg(\alpha)}\), the infimum being taken over all \(\alpha\) of degree \(\geq d\) lying with their conjugates outside \(V\), and obtain similar results.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 1770778
zbMATH Keywords: Lehmer constant
zbMATH Keywords: Mahler measure
zbMATH Keywords: polynomial zeros
author: A. Dubickas
author: Chris Smyth
MaRDI profile type: Publication
OpenAlex ID: W2042261493
cites work: Q4432295
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DOI: 10.5802/JTNB.330
description: scientific article; zbMATH DE number 1770778