Relations of linear dependence between the logarithms of algebraic numbers
Let \(\log\alpha_1,\dots, \log\alpha_m\) be logarithms of non-zero algebraic numbers and denote the rank over \(\mathbb{Z}\) of the \(\log\alpha_i\) by \(r\). The set of linear relations \({\mathcal L}= \{(t_1,\dots, t_m)\in\mathbb{Z}^m: \sum_{1\leq i\leq m}t_i\log\alpha_i= 0\}\) is a lattice of rank \(m-r\); denote its fundamental volume by \(H({\mathcal L})\). The author seeks upper bounds for \(H({\mathcal L})\) and proves that \[ H({\mathcal L})\leq (m- r+1)^{r/2} r^r(10D)^{3r}\Omega_r, \] where \(\Omega_r= \max_{1\leq j_1<\cdots< j_r\leq m} \prod_{1\leq i\leq m} V_j\), \(V_j= \max\left(h(\alpha_j), {|\log \alpha_j|\over D}\right)\), \(D=[\mathbb{Q}(\alpha_1,\dots, \alpha_m):\mathbb{Q}]\) and \(h\) denotes the absolute logarithmic height. From this bound and Minkowski's theory of successive minima, follow bounds for the size of a small base for \(\mathcal L\). Results of this type have important applications in diophantine approximation and in the theory of linear forms in logarithms. The result itself is an application of Minkowski's theorem in the geometry of numbers and the idea can be traced back to \textit{H. M. Stark} [Diophantine Approx. Appl., Proc. Conf. Washington 1972, 255-293 (1973; Zbl 0264.10023)]. As well as these precise estimates for linear forms in logarithms, there have been extensions to more general situations. For example, \textit{D. Bertrand} has obtained analogues for general algebraic groups [J. Aust. Math. Soc., Ser. A 62, 198-216 (1997; Zbl 0886.11035)].
scientific article; zbMATH DE number 1051567
scientific article