Lehmer's conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
type: Item
instance of: scholarly article
title: Lehmer's conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
zbMATH Open document ID: 1264.11049
author: Joseph H. Silverman
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2013-03-07T00:00:00Z
review text: Let \(h(\alpha)\) be the absolute logarithmic height, so that Lehmer's conjecture asserts that \newline \(\sum_{f(\alpha)=0} h(\alpha) \geq C>0\) for each monic polynomial \(f \in \mathbb{Z}[X]\) without roots of unity. The author proves that for all \(\varepsilon>0\) there is a constant \(C(\varepsilon)>0\) such that for each monic polynomial \(f \in \mathbb{Z}[X]\) without roots of unity, \(\deg f=D\), such that \(f(X)\) is divisible by \(X^{n-1}+\dots+X+1\) in the ring \((\mathbb{Z}/m \mathbb{Z})[X]\) for some integers \(m\geq 2\) and \(n\geq \max(\varepsilon D, 2)\) then \[ \sum_{f(\alpha)=0} h(\alpha) \geq C(\varepsilon) \log m. \] The proof uses some Fejér kernel estimate which in the context of Mahler measure was first used by \textit{P. E. Blanksby} and \textit{H. L. Montgomery} in 1971 [Acta Arith. 18, 355--369 (1971; Zbl 0221.12003)]. Some more precise conditions on a polynomial under which Lehmer's conjecture holds are also given. Finally, the author gives an elliptic analogue of this result.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11J97
zbMATH DE Number: 6142433
zbMATH Keywords: Lehmer conjecture
zbMATH Keywords: elliptic curve
zbMATH Keywords: canonical height
reviewed by: A. Dubickas
MaRDI profile type: Publication
Wikidata QID: Q122980801
OpenAlex ID: W2325465345
cites work: A lower bound for the height in abelian extensions
cites work: Algebraic integers near the unit circle
cites work: Lehmer's problem for polynomials with odd coefficients
cites work: THE MAHLER MEASURE OF POLYNOMIALS WITH ODD COEFFICIENTS
cites work: On a question of Lehmer and the number of irreducible factors of a polynomial
cites work: Auxiliary polynomials for some problems regarding Mahler's measure
cites work: The canonical height and integral points on elliptic curves
cites work: Q3972625
cites work: Q4486426
cites work: Lower bounds for heights in cyclotomic extensions
cites work: Q3309989
cites work: Q3994900
cites work: Q2770573
cites work: Q3671294
cites work: Factorization of certain cyclotomic functions
cites work: Counting points of small height on elliptic curves
cites work: The Weil height in terms of an auxiliary polynomial
cites work: Estimating heights using auxiliary functions
cites work: Q4317891
cites work: The Arithmetic of Elliptic Curves
cites work: On the Product of the Conjugates outside the unit circle of an Algebraic Integer
cites work: Some inequalities for certain power sums
cites work: The Mahler measure of algebraic numbers: a survey
cites work: Algebraic integers whose conjugates lie near the unit circle
DOI: 10.5802/JTNB.820
description: scientific article; zbMATH DE number 6142433