Inequalities on the Mahler measure of a polynomial
type: Item
instance of: scholarly article
title: Inequalities on the Mahler measure of a polynomial
zbMATH Open document ID: 0892.11035
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225375
full work available at URL: jtnb9-1.html
review text: The author proves the following inequalities for the Mahler measure \(M(P)\) which can be used to improve results of Bertin and Schinzel: Let \(P(x)\) be a monic polynomial with real coefficients, with all roots real and positive and with \(| P(1)| \geq 1\); then \[ M(P)^{1/d} \geq (1 + (4| P(0)| ^{1/d} + 1)^{1/2})/2. \] If instead, all roots are real and nonzero and \(| P(\pm 1)| \geq 1\) then \[ M(P)^{2/d} \geq (1 + (4| P(0)| ^{2/d} + 1)^{1/2})/2. \] The proofs use the auxiliary function \(f(x) = \log^+(x) - (c\log| x| + (1 - 2c)\log| x-1| )\). Some related inequalities established by similar techniques may be found in the author's article [\textit{V. Flammang}, Can. Math. Bull. 38, 438-444 (1995; Zbl 0842.26011)].
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 12E05
zbMATH DE Number: 1132888
zbMATH Keywords: polynomial
zbMATH Keywords: Mahler measure
zbMATH Keywords: inequality
author: Valérie Flammang
reviewed by: David W. Boyd
MaRDI profile type: Publication
OpenAlex ID: W2327076138
cites work: On the product of the conjugates outside the unit circle of an algebraic number
DOI: 10.5802/JTNB.190
description: scientific article; zbMATH DE number 1132888