Effective estimates for generalized global relations. (English. Russian original) Zbl 1176.11035

Azamatov, T. R.

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)].

Reviewer: Wadim Zudilin (Bonn)

MSC:

11J91	Transcendence theory of other special functions
11J61	Approximation in non-Archimedean valuations

Keywords:

algebraic independence; non-Archimedean valuations; global relation; arithmetic special function