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On estimates for the orders of zeros of polynomials in analytic functions and their application to estimates of the relative transcendence measure of values of \(E\)-functions. (Russian) Zbl 0516.10025

The first part of this paper is an extension of work of Nesterenko on the order of the zeros of polynomials \(P(z,f_1,\ldots, f_m)\), where \(f_1,\ldots, f_m\) satisfy a system of first order linear differential equations. Similar results have been obtained by W. D. Brownawell and D. W. Masser [Duke Math. J. 47, 273–295 (1980; Zbl. 461.10027)]. The second part uses these results to obtain algebraic independence measures for the values at algebraic points of \(E\)-functions defined over the rationals or an imaginary quadratic field. The results are similar to, but more refined than, those given by A. B. Shidlovskiĭ [J. Aust. Math. Soc., Ser. A 27, 385–407 (1979; Zbl. 405.10024)].

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J91 Transcendence theory of other special functions
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