The maximal conjugate of a non-reciprocal algebraic integer
type: Item
instance of: scholarly article
title: The maximal conjugate of a non-reciprocal algebraic integer
zbMATH Open document ID: 0890.11030
DOI: 10.1007/BF02465885
published in: Lithuanian Mathematical Journal
publication date: 1998-03-26T00:00:00Z
review text: For an algebraic integer \(a\) denote by \(\overline{|a|}\) the maximal modulus of conjugates of \(a\). Assume that \(a\) is non-reciprocal, i.e. its minimal polynomial \(P\) does not satisfy the identity \(P(x)=x^nP(1/x)\), with \(n\) denoting the degree of \(a\). \textit{J. W. S. Cassels} [J. Math. Sci. 1, 1-8 (1966; Zbl 0147.30803)] proved the inequality \(\overline{|a|}>1+0.1/n\), and from a later result of \textit{C. J. Smyth} [Bull. Lond. Math. Soc. 3, 169-175 (1971; Zbl 0235.12003)] the stronger inequality \(\overline{|a|}>1+c/n\) with \(c=0.2811\dots\) follows. The author shows that for \(n>n_0(\varepsilon)\) (with effective \(n_0(\varepsilon)\)) one can take \(c=\omega-\varepsilon\) where \(\omega=0.3096\dots\) is defined as a root of a rather complicated transcendental equation.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 1131034
zbMATH Keywords: size of algebraic integers
zbMATH Keywords: non-reciprocal polynomials
zbMATH Keywords: maximal conjugate
author: A. Dubickas
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02465885
OpenAlex ID: W1975502193
description: scientific article; zbMATH DE number 1131034