Products and quotients of linear forms of logarithms of algebraic numbers: conjectures and partial results
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)].
scientific article; zbMATH DE number 5302780