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Linear forms in two logarithms and Schneider’s method. III. (English) Zbl 0702.11044
Verf. verfeinern ihre in [Acta Arith. 53, No.3, 251-287 (1989; Zbl 0642.10034)] erhaltene untere Abschätzung für \(| b_ 1 \log \alpha_ 1-b_ 2 \log \alpha_ 2| \neq 0\) bei algebraischen \(\alpha_ j\neq 0\) und ganzrationalen \(b_ j\). Dazu kombinieren sie ihre a.a.O. entwickelte Methode mit einer Technik, die sie bereits in [Math. Ann. 231, 241-267 (1978; Zbl 0349.10029)] vorgestellt haben. Die numerischen Resultate sind jetzt erheblich verbessert, was für eine Reihe von Anwendungen von großer Bedeutung ist, z.B. bei Problem der Klassenzahl 1 imaginär-quadratischer Zahlkörper oder bei exponentiellen diophantischen Gleichungen des Typs \(a^ x-b^ y=c.\) Gesondert bearbeitet ist schließlich der Spezialfall, wo ein \(\alpha_ j\) eine Einheitswurzel ist.
Reviewer: P.Bundschuh

MSC:
11J86 Linear forms in logarithms; Baker’s method
11J04 Homogeneous approximation to one number
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References:
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