Products and quotients of linear forms of logarithms of algebraic numbers: conjectures and partial results
type: Item
instance of: scholarly article
title: Products and quotients of linear forms of logarithms of algebraic numbers: conjectures and partial results
zbMATH Open document ID: 1163.11055
author: Guy Diaz
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/55503
review text: The strong four exponentials Conjecture deals with the \(\bar\mathbb Q\)-vector space \(\widetilde{\mathcal L}\) spanned by \(1\) and by all the logarithms of the non-zero algebraic numbers: it states that if \(x_1\), \(x_2\) are \(\bar\mathbb Q\)-linearly independent complex numbers and if \(y_1\), \(y_2\) are \(\bar\mathbb Q\)-linearly independent complex numbers, then one at least of the four numbers \(x_1y_1\), \(x_1y_2\), \(x_2y_1\), \(x_2y_2\) does not belong to \(\widetilde{\mathcal L}\). The author proposes a number of conjectures which are either equivalent or else consequences of the four exponentials Conjecture. He also produces new consequences of the strong six exponentials Theorem of \textit{D.~Roy} [ J. Number Theory 41, No. 1, 22--47 (1992; Zbl 0763.11030)] which yield special cases of the above-mentioned conjecture. A typical example is as follows: the strong four exponentials Conjecture holds under the extra assumption that the three numbers \(y_1\), \(y_2\), \(1/x_1\) are \(\bar\mathbb Q\)-linearly independent and \(x_2/x_1\in\widetilde{\mathcal L}\). Related works by the author are [ J. Théor. Nombres Bordx. 9, No. 1, 229--245 (1997; Zbl 0887.11030) and J. Théor. Nombres Bordx. 16, No. 3, 535--553 (2004; Zbl 1071.11046)]. See also [\textit{M. Waldschmidt}, Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften. 326. Berlin: Springer (2000; Zbl 0944.11024)].
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5302780
zbMATH Keywords: four exponentials conjecture
zbMATH Keywords: transcendence
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: Schanuel's Conjecture
MaRDI profile type: Publication
OpenAlex ID: W2312991798
Wikidata QID: Q123111119
cites work: The four exponentials conjecture and D. Bertrand's conjecture on the modular function
cites work: Use of complex conjugation in the transcendence of values of the usual exponential function
cites work: Matrices whose coefficients are linear forms in logarithms
cites work: Q4488162
cites work: Q5703071
cites work: Q5486693
DOI: 10.5802/JTNB.592
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description: scientific article; zbMATH DE number 5302780