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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 $$P(z,f_ 1(z),\dots,f_ s(z))\not\equiv 0$$.

##### MSC:
 11J91 Transcendence theory of other special functions