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On approximation, by algebraic numbers, of the solutions of some equations containing $$E$$-functions. (English. Russian original) Zbl 0896.11028
Proc. Steklov Inst. Math. 207, 63-66 (1995); translation from Tr. Mat. Inst. Steklova 207, 66-69 (1994).
Let $$K$$ be an algebraic number field of finite degree, and let $$f_1(z), \dots, f_s(z)$$ be a set of $$E$$-functions with coefficients in $$K$$. Suppose that these functions comprise a solution of the system of linear differential equations $y_i'= Q_{i0} (z)+ \sum^s_{j=1} Q_{ij} (z)y_j, \quad Q_{ij} (z)\in \mathbb{C}(z), \quad i=1, \dots,s.$ The author states a new result without proof:
If $$\xi\in \mathbb{C}$$ satisfies the equation $$P(z,f_1(z), \dots, f_s(z))=0$$, where $$P(x_0,x_1, \dots, x_s)$$ is a polynomial with algebraic coefficients such that $$P(z,f_1(z), \dots,f_s(z)) \not\equiv 0$$, then for any $$\varepsilon>0$$ and $$\kappa>0$$, the system of inequalities $|\xi- \theta| <\exp \biggl(-\bigl( h(\theta) \bigr)^\varepsilon \biggr), \quad \kappa (\theta)\leq \kappa$ has only finitely many solutions in algebraic numbers $$\theta$$, where $$\kappa (\theta)$$ and $$h(\theta)$$ denote the degree and height of $$\theta$$ respectively. This theorem allows us to obtain more easily the algebraic independence of $$\xi$$ and certain numbers connected with $$f_1(z), \dots, f_s(z)$$.
For the entire collection see [Zbl 0833.00028].
##### MSC:
 11J17 Approximation by numbers from a fixed field 11J85 Algebraic independence; Gel’fond’s method 11J91 Transcendence theory of other special functions 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)