Estimates for measures of algebraic independence of values of \(E\)- functions (Q1333642)

Let \(f_ 1(z),\dots, f_ m(z)\), belonging to the class \(\mathbb{K} E\), satisfy a system of differential equations and be uniformly algebraically dependent. Suppose that the totality of products of the powers \(f_ 1^{k_ 1}(z) \dots f_ m^{k_ m} (z)\), \(\sum_{i=1}^ m k_ i=N\), for any natural number \(N\), composes a non-reducible system. The author proves some effective estimates for measures of algebraic independence of values of parts of \(f_ 1(z), \dots, f_ m(z)\) in algebraic points. These results are certain developments of earlier works by the same author [Sib. Math. J. 31, No. 5, 732-743 (1990); translation from Sib. Mat. Zh. 31, No. 5, 31-45 (1990; Zbl 0738.11044)].