On estimates for the orders of zeros of polynomials in analytic functions and their application to estimates for the relative transcendence measure of values of $E$-functions

This paper gives estimates for the orders of zeros of polynomials in a set of analytic functions satisfying a system of linear differential equations with coefficients in $\mathbf C(z)$, in the case when these functions are algebraically dependent over $\mathbf C(z)$. Using the Siegel–Shidlovskii method, these estimates are applied to obtain effective bounds from below for the relative transcendence measure of the values of $E$-functions in the case when the basic set of $E$-functions is algebraically dependent over $\mathbf C(z)$.
Bibliography: 20 titles.