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On solutions of some equations involving E-functions. (Russian) Zbl 0767.11031
Let \(f_ 1(z),\dots,f_ s(z)\) be a set of \(E\)-functions constituting a solution of a system of linear differential equations \[ y_ i'=q_{i0}(z)+ \sum_{j=1}^ s q_{ij}(z)y_ j, \qquad i=1,\dots,s, \quad q_{ij}(z)\in\mathbb{C}(z), \] and let \(\zeta\) be a complex number such that \(P(\zeta,f_ 1(\zeta),\dots,f_ s(\zeta))=0\), where \(P(x_ 0,x_ 1,\dots,x_ s)\) is a polynomial with algebraic coefficients satisfying \(P(z,f_ 1(z),\dots,f_ s(z))\not\equiv 0\). The author proves that for any \(\varepsilon>0\) and \(\kappa>0\) the set of inequalities \[ |\zeta- \theta| <\exp(-(h(\theta))^ \varepsilon), \qquad \deg\theta\leq\kappa \] has a finite number of solutions in algebraic \(\theta\), where \(h(\theta)\) and \(\deg\theta\) are the height and the degree of \(\theta\), respectively. From this theorem we can deduce certain results about the algebraic independence of the values of \(E\)-functions at algebraic points under a weaker condition, that is, the usual algebraic independence assumption for \(E\)-functions \(f_ 1,\dots,f_ s\) is replaced by \(P(z,f_ 1(z),\dots,f_ s(z))\not\equiv 0\).

11J91 Transcendence theory of other special functions