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Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques. (New methods for lower bounds of linear forms in logarithms of algebraic numbers). (French) Zbl 0733.11020

Bisher war Bakers Methode im Fall \(n\geq 3\) die einzige, die zu effektiven unteren Schranken für nichttriviale Linearformen \(\beta_ 1 \log \alpha_ 1+...+\beta_ n \log \alpha_ n\) in Logarithmen algebraischer \(\alpha_ 1,...,\alpha_ n\) führte. Während jedoch Bakers Ansatz Gel’fonds Lösung des siebten Hilbert-Problems verallgemeinerte, basiert der Ansatz des Verf. auf Schneiders Lösung des genannten Problems. Verf. benutzt dabei weder eine Extrapolationstechnik noch Kummertheorie; auch sind seine Hilfsfunktionen gänzlich von den Bakerschen verschieden. Seine Methode ist dual (im Sinne seiner Arbeit [J. Anal. Math. 56, 231-254, 255-279 (1991)]) zu der von N. Hirata-Kohno [Sémin. Théor. Nombres Paris 1988-89, Prog. Math. 91, 117-140 (1990; Zbl 0716.11033)] im elliptischen Fall angewandten und führt in der hier behandelten Situation \(\beta_ 1,...,\beta_ n\in {\mathbb{Z}}\) zu einem völlig expliziten Resultat (Théorème 1.1), das quantitativ dieselbe Güte erreicht, wie sie Bakers Methode heute herzuleiten gestattet.

MSC:

11J86 Linear forms in logarithms; Baker’s method
11J82 Measures of irrationality and of transcendence

Citations:

Zbl 0716.11033
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References:

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