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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)].
##### MSC:
 11J91 Transcendence theory of other special functions 11J61 Approximation in non-Archimedean valuations
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