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

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.


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