On algebraic numbers of small measure
type: Item
instance of: scholarly article
title: On algebraic numbers of small measure
zbMATH Open document ID: 0860.11063
DOI: 10.1007/BF02348822
published in: Lithuanian Mathematical Journal
publication date: 1997-04-24T00:00:00Z
review text: For a nonzero algebraic number \(\alpha\) of degree \(d\) and conjugates \(\alpha_1 = \alpha\), \(\alpha_2, \dots, \alpha_d\), denote by \(\house\alpha\) the maximum of the \(\alpha_i\)'s and by \(M(\alpha)\) its Mahler measure, i.e. the product \(q \prod^d_{i=1} \max \{1,|\alpha_i |\}\). It is shown that if \(\alpha\) is not a root of unity then for sufficiently large \(d\) one has \[ |\alpha-1 |> \exp \biggl(-(\pi/4 + \varepsilon) \bigl(d \log d \log M (\alpha) \bigr)^{1/2} \biggr) \] which improves a result of \textit{Y. Bugeaud}, \textit{M. Mignotte} and \textit{F. Normandin} [C. R. Acad. Sci., Paris, Sér. I 321, No. 5, 517-522 (1995; Zbl 0848.11047)] who had \((2/3)^{1/2}\) in place of \(\pi/4\). Moreover an upper bound for the number of real zeros of an irreducible polynomial over \(\mathbb{Z}\) is obtained in terms of the Mahler measure of its root and this is used to obtain for sufficiently large and all totally real \(\alpha\) of degree \(d\) and not of the form \(2\cos (\pi r)\) with rational \(r\) the inequality \[ \house\alpha > 2+3.8 {\log \log^3d \over d \log^4d}, \] which essentially improves a previous result of the author [Acta Arith. 63, 15-20 (1993; Zbl 0777.11039)].
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 939538
zbMATH Keywords: Mahler measure
zbMATH Keywords: upper bound for the number of real zeros of an irreducible polynomial
author: A. Dubickas
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
cites work: Q4864317
cites work: On a conjecture of A. Schinzel and H. Zassenhaus
cites work: Q3222282
cites work: On algebraic numbers of small height: Linear forms in one logarithm
cites work: A refinement of two theorems of Kronecker
full work available at URL: https://doi.org/10.1007/bf02348822
OpenAlex ID: W4236908706
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description: scientific article; zbMATH DE number 939538