Matrices whose coefficients are linear forms in logarithms. (English) Zbl 0763.11030

Roy, Damien

J. Number Theory 41, No. 1, 22-47 (1992).

Denote by \(L\) the \({\mathbb{Q}}\)-vector space of complex numbers \(\ell\) such that \(e^{\ell}\) is an algebraic number, and by \({\mathcal L}\) the vector space generated by \(1\) and \(L\) over the field \(\overline\mathbb{Q}\) of algebraic numbers. The elements of \(L\) are the logarithms of non-zero algebraic numbers, while the elements of \({\mathcal L}\) are the linear forms in logarithms of algebraic numbers: \(\beta_{0}+\beta_{1}\log\alpha_{1}+\cdots+ \beta_{n}\log\alpha_{n}\). One result of this paper is a lower bound for the rank of a matrix whose entries are in \({\mathcal L}\).
The simplest case is as follows: a \(2\times3\) matrix with entries in \({\mathcal L}\) whose columns are \(\overline\mathbb{Q}\)-linearly independent, and whose rows are \({\overline\mathbb{Q}}\)-linearly independent, has rank 2 (the six exponentials theorem deals with a \(2\times3\) matrix with entries in \(L\)). The proof relies on the theorem of the linear subgroup, which is sharpened by the author; this refinement is performed by means of arguments where the author uses the language of category theory. He deduces improvements of earlier results due to M. Emsalem [C. R. Acad. Sci., Paris, Sér. I 297, 225-227 (1983; Zbl 0529.12006); J. Reine Angew. Math. 382, 181-198 (1987; Zbl 0621.12008)]and M. Laurent [J. Reine Angew. Math. 399, 81-108 (1989; Zbl 0666.12001)]on the conjectures of Leopoldt and Jaulent concerning the \(p\)-adic rank of \(S\)- units in a number field.

Reviewer: M.Waldschmidt (Paris)

MSC:

11J81	Transcendence (general theory)
11R27	Units and factorization

Keywords:

logarithms of non-zero algebraic numbers; linear forms in logarithms of algebraic numbers; lower bound for the rank of a matrix; \(p\)-adic rank of \(S\)-units; Leopoldt conjecture; Jaulent conjecture

Citations:

Zbl 0529.12006; Zbl 0621.12008; Zbl 0666.12001

References:

[1]	Baker, A., (Transcendental Number Theory (1975), Cambridge Univ. Press: Cambridge Univ. Press London) · Zbl 0297.10013
[2]	Bourbaki, N., Algèbre (1970), Diffusion CCLS: Diffusion CCLS Paris, Chap. 2 · Zbl 0211.02401
[3]	Brumer, A., On the units of algebraic number fields, Mathematika, 14, 121-124 (1967) · Zbl 0171.01105
[4]	Emsalem, M., Rang \(p\)-adique de groupes de \(S\)-unités d’un corps de nombres, C. R. Acad. Sci. Paris Sér. I, 297, 225-227 (1983) · Zbl 0529.12006
[5]	Emsalem, M., Sur les idéaux dont l’image par l’application d’Artin dans une \(Z_p\)-extension est triviale, J. Reine Angew. Math., 382, 181-198 (1987) · Zbl 0621.12008
[6]	Jaulent, J.-F, Sur l’indépendance \(l\)-adique de nombres algébriques, J. Number Theory, 20, 149-158 (1985) · Zbl 0571.12007
[7]	Laurent, M., Rang \(p\)-adique d’unités et action de groupes, J. Reine Angew. Math., 399, 81-108 (1989) · Zbl 0666.12001
[8]	MacLane, S., (Categories for the Working Mathematician (1971), Springer-Verlag: Springer-Verlag New York) · Zbl 0232.18001
[9]	D. Roy; D. Roy
[10]	Waldschmidt, M., Transcendance et exponentielles en plusieurs variables, Invent. Math., 63, 97-127 (1981) · Zbl 0454.10020
[11]	Waldschmidt, M., Dependence of logarithms of algebraic points, Colloq. Math. Soc. János Bolyai, 51, 1013-1035 (1987) · Zbl 0714.11043
[12]	Waldschmidt, M., On the transcendence methods of Gel’fond and Schneider in several variables, (New Advances in Transcendence Theory (1988), Cambridge Univ. Press: Cambridge Univ. Press New York/London), 375-398 · Zbl 0659.10035