Products and quotients of linear forms of logarithms of algebraic numbers: conjectures and partial results. (Produits et quotients de combinaisons linéaires de logarithmes de nombres algébriques: conjectures et résultats partiels.) (French. English summary) Zbl 1163.11055

Diaz, Guy

J. Théor. Nombres Bordx. 19, No. 2, 373-391 (2007).

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 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 [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)].

Reviewer: Michel Waldschmidt (Paris)

MSC:

11J81	Transcendence (general theory)
11J86	Linear forms in logarithms; Baker’s method

Keywords:

four exponentials conjecture; transcendence; linear forms in logarithms; Schanuel’s Conjecture

Citations:

Zbl 0763.11030; Zbl 0887.11030; Zbl 1071.11046; Zbl 0944.11024

References:

[1]	G. Diaz, La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire. J. Théorie des Nombres de Bordeaux 9 (1997), 229-245. · Zbl 0887.11030
[2]	G. Diaz, Utilisation de la conjugaison complexe dans l’étude de la transcendance de valeurs de la fonction exponentielle usuelle. J. Théorie des Nombres de Bordeaux 16 (2004), 535-553. · Zbl 1071.11046
[3]	D. Roy, Matrices whose coefficients are linear forms in logarithms. J. Number Theory 41 (1992), 22-47. · Zbl 0763.11030
[4]	M. Waldschmidt, Diophantine approximation on linear algebraic groups. Grundlerhen der Mathematischen Wissenschaften 326, Springer-Verlag, 2000. · Zbl 0944.11024
[5]	M. Waldschmidt, Variations on the six exponentials theorem. Algebra and Number Theory, Proceedings of the Silver Jubilee Conference University of Hyderabad, ed R.Tandon, Hindustant Book Agency, 2005, 338-355. · Zbl 1176.11033
[6]	M. Waldschmidt, Further variations on the six exponentials theorem. Hardy-Ramanujan J. 28 (2005), 1-9. · Zbl 1116.11054