Lower bounds for heights in cyclotomic extensions
type: Item
instance of: scholarly article
title: Lower bounds for heights in cyclotomic extensions
zbMATH Open document ID: 1203.11072
author: M. I. M. Ishak
author: Michael J. Mossinghoff
author: Benjamin Wiles
author: Chris Pinner
published in: Journal of Number Theory
publication date: 2010-06-18T00:00:00Z
review text: Let \(f(x) = a_{0} \prod_{i=1}^{d} (x-\alpha_{i}) = a_{0}x^{n}+a_{1}x^{n-1}+ \cdots + a_{n}\) be an irreducible polynomial with integer coefficients. The Mahler measure of \(f\) is defined to be \(M(f) =\prod_{i=1}^{d}\max \{ 1, |\alpha_{i}| \}\). If \(\alpha \neq 0\) is a root of \(f\), then the absolute logarithmic height \(h(\alpha)\) of \(\alpha\) is defined by \(h(\alpha) = \frac{\log M(f)}{d}\). Lehmer's problem asks if there exists a positive constant \(c>0\) such that \(h(\alpha)> c/d\) as long as \(\alpha\) is neither zero nor a root of unity. The main result of the paper under review is that if \(\alpha\) is a nonzero element of a finite abelian extension of the rationals and is not equal to a root of unity, then \(h(\alpha) \geq 0.155097\). The authors also consider several special cases in which improved lower bounds are given.
reviewed by: Henri Johnston
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5723357
zbMATH Keywords: heights
zbMATH Keywords: Mahler measure
zbMATH Keywords: Lehmer's problem
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.01.014
OpenAlex ID: W2017107324
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DOI: 10.1016/J.JNT.2010.01.014
description: scientific article; zbMATH DE number 5723357