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The Hilbert polynomial and linear forms in the logarithms of algebraic numbers. (English. Russian original) Zbl 1214.11088
Izv. Math. 72, No. 6, 1063-1110 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 6, 5-52 (2008).
Until recently, the best known estimates for lower bounds of linear forms in logarithms of algebraic numbers were due to A. Baker and G. Wüstholz on the one hand [“Logarithmic forms and group varieties”, J. Reine Angew. Math. 442, 19–62 (1993; Zbl 0788.11026)], using Baker’s method, and to the reviewer on the other hand [M. Waldschmidt, “Minorations de combinaisons linéaires de logarithmes de nombres algébriques”, Can. J. Math. 45, No. 1, 176–224 (1993; Zbl 0774.11036)], by means of an extension of Schneider’s method to several variables. See [A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry. New Mathematical Monographs 9. Cambridge: Cambridge University Press (2007; Zbl 1145.11004)] and [M. Waldschmidt, Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften. 326. Berlin: Springer (2000; Zbl 0944.11024)] for further references.
In 1998 and 2000, E. M. Matveev refined these estimates by means of arguments from geometry of numbers [see “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II”, Izv. Math. 64, No. 6, 1217–1269 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 6, 125–180 (2000; Zbl 1013.11043) and “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers”, Izv. Math. 62, No. 4, 723–772 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 81–136 (1998; Zbl 0923.11107); see also Yu. Nesterenko, “Linear forms in logarithms of rational numbers”, Lect. Notes Math. 1819, 53–106 (2003; Zbl 1044.11069)].
Here, the author introduces new ideas involving Hilbert polynomials and reaches sharper lower bounds which are likely to be used from now on in the proofs of Diophantine results which rest on such an estimate.

MSC:
11J86 Linear forms in logarithms; Baker’s method
11J25 Diophantine inequalities
11H06 Lattices and convex bodies (number-theoretic aspects)
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