Linear forms in the logarithms of three positive rational numbers
type: Item
instance of: scholarly article
title: Linear forms in the logarithms of three positive rational numbers
zbMATH Open document ID: 0905.11032
author: Curtis D. Bennett
author: Josef Blass
author: A. M. W. Glass
author: David B. Meronk
author: Ray Phillip Steiner
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/93689
full work available at URL: jtnb9-1.html
review text: Let \(\alpha_{1},\alpha_{2},\alpha_{3}\) be multiplicatively independent rational numbers, each \(>1\); let \(b_{1},b_{2},b_{3}\) be positive integers with \(b_{2}\log\alpha_{2}>\max\{b_{1}\log\alpha_{1},b_{3}\log\alpha_{ 3}\}\), such that the number \(\Lambda=b_{2}\log\alpha_{2}-b_{1}\log\alpha_{1}-b_{3}\log\alpha_{3}\) does not vanish. Write \(\alpha_{j}=\alpha_{j1}/\alpha_{j2}\) and \(a_{j}=\max\{\alpha_{j1},e\}\) \ (\(j=1,2\)). Further, define \[ b'=\left({b_{2}\over\log a_{1}}+{b_{1}\over\log a_{2}}\right) \left({b_{2}\over\log a_{3}}+{b_{3}\over\log a_{2}}\right) \] and \(B=\max\{10,\log b'\}\). The conclusion of Theorem 1 is then \[ | \Lambda| \geq\exp\{-4.5\cdot 10^{5} B^{2}\log a_{1}\log a_{2}\log a_{3}\}. \] There is however a further technical hypothesis, namely that \(\gcd(b_{1},b_{2},b_{3})=1\) and \(d_{1}b_{1}+d_{2}b_{2}+d_{3}b_{3}\not=0\) for any triple \((d_{1},d_{2},d_{3})\not=(0,0,0)\) of rational integers with either \[ 0<| d_{2}| \leq 10^{4}B\log a_{2}\min\{\log a_{1},\log a_{3}\} \] and \[ \max\{| d_{1}| , | d_{3}| \}\leq 10^{4}B\log a_{1}\log a_{3} \] or else \(d_{2}=0\) and \[ | d_{i}| \leq 10^{4}B\log a_{2} \log a_{i}, \quad i=1,3. \] Theorem 2 of the paper under review provides a refined estimate with better numerical values than the above \(4.5\cdot 10^{5}\) and \(10^{4}\) respectively. The proof is an extension in two variables of the work on linear forms in two logarithms by \textit{M. Laurent, M. Mignotte} and \textit{Yu. V. Nesterenko} [J. Number Theory 55, No. 2, 285-321 (1995; Zbl 0843.11036)] using Laurent's interpolation determinants [\textit{M. Laurent}, Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 209-230 (1991; Zbl 0762.11027); Acta Arith. 66, No. 2, 181-199 (1994; Zbl 0801.11034); Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/Fr. 1990, 215-238 (1992; (Zbl 0773.11047)]. Their zero estimate involves a refinement of that of \textit{P. Philippon} [Bull. Soc. Math. Fr. 114,…
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 1132890
zbMATH Keywords: interpolation determinants
zbMATH Keywords: linear forms in three logarithms
zbMATH Keywords: zero estimates
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
OpenAlex ID: W2029220592
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DOI: 10.5802/JTNB.192
description: scientific article; zbMATH DE number 1132890