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Linear forms in two logarithms and Schneider’s method. II. (English) Zbl 0642.10034

Let \(\alpha _ 1,\alpha _ 2\) be two multiplicatively independent algebraic numbers, and log \(\alpha\) \({}_ 1\), log \(\alpha\) \({}_ 2\) two non-zero determinations of their logarithms. Denote by D the degree of the field \({\mathbb{Q}}(\alpha _ 1,\alpha _ 2)\) over \({\mathbb{Q}}\). Further, let \(b_ 1,b_ 2\) be two positive rational integers such that \(b_ 1 \log \alpha _ 1\neq b_ 2 \log \alpha _ 2\). Define \(B=\max \{b_ 1,b_ 2\}\), and choose two positive real numbers \(a_ 1,a_ 2\) satisfying \[ a_ j\geq 1,a_ j\geq h(\alpha _ j)+\log 2,\quad a_ j\geq \frac{2e}{D}\cdot \log \alpha _ j \] for \(j=1\) and \(j=2\). Here, h denotes the absolute logarithmic height. Then \[ | b_ 1 \log \alpha _ 1-b_ 2 \log \alpha _ 2| \geq \exp \{-500\cdot D^ 4a_ 1a_ 2(7.5+\log B)^ 2\}. \] This is a refinement of the rational case in the result of the authors [Math. Ann. 231, 241-267 (1978; Zbl 0349.10029)], and of the case of two logarithms in the result of J. Blass, A. M. W. Glass, D. K. Manski, D. B. Meronk and R. P. Steiner [Constants for lower bounds for linear forms in the logarithms of algebraic numbers, Acta Arith. (to appear)].
Reviewer: M.Waldschmidt

MSC:

11J81 Transcendence (general theory)
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