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Zbl 1010.11036
Bilu, Yuri; Bugeaud, Yann
Proof of the Baker-Feldman theorem via linear forms in two logarithms. (Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes.) (French)
[J] J. Théor. Nombres Bordx. 12, No. 1, 13-23 (2000). ISSN 1246-7405

{\it A. O. Gel'fond} [Transcendental and algebraic numbers. New York: Dover Publications (1960; Zbl 0090.26103)] proved effective measures of linear independence for two logarithms of algebraic numbers; he also deduced a non effective measure of linear independence for $n$ logarithms of algebraic numbers from Thue's improvement of Liouville's inequality. The first effective measure of linear independence for $n$ logarithms of algebraic numbers was produced by A. Baker. By means of a refinement of Baker's inequality, N. I. Feldman obtained an effective refinement of Liouville's inequality as follows: For each real algebraic number $\alpha$ of degree $d\ge 3$ there exist two effective positive constants $C$ and $\tau$ such that, for any $p/q\in\bbfQ$ with $q>0$, $$ \left|\alpha-{p\over q}\right|>Cq^{-\tau}. $$ Using an argument akin to Gel'fond's one, E. Bombieri pointed out that measures of linear independence for two logarithms suffice to deduce measures of linear independence for $n$ logarithms, which are good enough to entail Feldman's improved Liouville's inequality. He used it both in the archimedean case [{\it E. Bombieri}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, 61--89 (1993; Zbl 0774.11034)] and in the ultrametric case, where Laurent's interpolation determinants was used [{\it E. Bombieri} and {\it P. B. Cohen}, ibid. 24, 205--225 (1997; Zbl 0912.11028)].\par In the paper under review, in place of using Dyson's type methods, the authors apply Schneider's transcendence method which produces Baker-like estimates. Further, a complete proof of the required measure of linear independence for two logarithms is also included, by means of Laurent's interpolation argument. As a result this paper is self-contained.
[Michel Waldschmidt (Paris)]

MSC 2000:: *11J68 Approximation to algebraic numbers
11J86 Linear forms in logarithms; Baker's method

Keywords: measure of linear independence; logarithms of algebraic numbers; measures of linear independence for two logarithms; Schneider's transcendence method

Citations: Zbl 0090.26103; Zbl 0774.11034; Zbl 0912.11028

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