Heights of roots of polynomials with odd coefficients
type: Item
instance of: scholarly article
title: Heights of roots of polynomials with odd coefficients
zbMATH Open document ID: 1239.11113
author: Juan-Miguel Gracia
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2011-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/116409
review text: For an integer \(m\geq 2\) let \(D_m\) denote the set of integer polynomials whose coefficients are all \(\equiv 1\pmod m\). The authors prove: If \(\alpha\) is a zero of a polynomial \(f\) in \(D_m\) of degree \(n\) which is not a \( 2(n+1)\) root of unity (not a \((n+1)\) root of unity if \(m\geq 3\)), then the height of \(\alpha\) satisfies \[ h(\alpha)\geq \frac{c_m}{n+1} \] with \[ c_2=0.427800 \] and \[ c_m=\log\left(\frac m2\right)+\frac{2.947486-\frac {\delta}{2}}{m^2}+O\left(\frac{1}{ m^4}\right), \] where \(\delta=1\) if \(m\geq 3\) odd, \(\delta=0\) if \(m\geq 4\) even. The optimal \(c_m\) satisfies \(c_m=\log m+O(1)\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11R06
zbMATH DE Number: 5862105
zbMATH Keywords: polynomials with odd coefficients
zbMATH Keywords: roots
zbMATH Keywords: height
MaRDI profile type: Publication
OpenAlex ID: W2324677408
cites work: Lehmer's problem for polynomials with odd coefficients
cites work: Auxiliary polynomials for some problems regarding Mahler's measure
cites work: Lower bounds for heights in cyclotomic extensions
cites work: Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante
DOI: 10.5802/JTNB.721
description: scientific article; zbMATH DE number 5862105