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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].
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)