Nguyen Tien Tai 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 Mat. Sb., N. Ser. 120(162), No. 1, 112-142 (1983). 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)]. Reviewer: John H. Loxton (Greenwich) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 2 Documents MSC: 11J85 Algebraic independence; Gel’fond’s method 11J91 Transcendence theory of other special functions Keywords:order of zeros of polynomials; system of first order linear differential equations; algebraic independence measures; values at algebraic points of E-functions Citations:Zbl 0461.10027; Zbl 0405.10024 PDFBibTeX XMLCite \textit{Nguyen Tien Tai}, Mat. Sb., Nov. Ser. 120(162), No. 1, 112--142 (1983; Zbl 0516.10025) Full Text: MNR