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Effective estimates for generalized global relations. (English. Russian original) Zbl 1176.11035
Russ. Math. Surv. 62, No. 5, 980-982 (2007); translation from Usp. Mat. Nauk 62, No. 5, 145-146 (2007).
The author announces a general theorem on the quantitative algebraic independence of the values of so-called \(F\)-functions (in the sense of [V. G. Chirskii, Math. Notes 48, No. 2, 795–798 (1990); translation from Mat. Zametki 48, No. 2, 123–127 (1990; Zbl 0764.11031)]) satisfying a system of linear differential equations with coefficients from \(\overline{\mathbb Q}(z)\), at transcendental points admitting sufficiently good approximations by algebraic numbers. This theorem may be thought of as an effective solution to the problem on the nonexistence of global relations for the \(F\)-series in question; [cf. D. Bertrand, V. Chirskii, and J. Yebbou, Ann. Fac. Sci. Toulouse, Math. (6) 13, No. 2, 241–260 (2004; Zbl 1176.11036)].
11J91 Transcendence theory of other special functions
11J61 Approximation in non-Archimedean valuations
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