## 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)

### Citations:

Zbl 0349.10029; Zbl 0361.10027
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