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Lower bounds for polynomials on values of algebraically dependent E-functions. (Russian. English summary) Zbl 0876.11038
Let \({\mathbb{I}}\) be an imaginary quadratic field and \({\mathbb{K}}\) be an algebraic number field of finite degree over \({\mathbb{I}}\). Suppose that the \({\mathbb{K}}\)E-functions \(f_1(z)\), \(f_2(z), \dots, f_s(z)\) form a solution of the system of linear differential equations \[ y_i'= Q_{i0} (z)+ \sum^s_{j=1} Q_{ij} (z)y_j,\;i=1, \dots, s,\;Q_{ij} (z)\in {\mathbb{K}} (z), \] and that \(\alpha\) is an algebraic number, distinct from the zeros and the poles of all the coefficients \(Q_{ij} (z)\). The author proves that for any polynomial \(P(x_1, \dots, x_s)\in \mathbb{Z}_{\mathbb{I}} [x_1, \dots, x_s]\) of degree \(d\) and height \(H\) (here \(\mathbb{Z}_{\mathbb{I}}\) is the ring of algebraic integers in \({\mathbb{I}})\), either \[ P\bigl(f_1(\alpha), \dots, f_s(\alpha) \bigr)=0 \quad\text{or}\quad \biggl|P\bigl(f_1 (\alpha), \dots, f_s(\alpha) \bigr)\biggr |> CH^{-\gamma \kappa^{\ell+ 1} d^\ell}, \] where \(\kappa =[{\mathbb{K}} (\alpha): {\mathbb{I}}]\geq 2\), and \(\ell\) is the number of functions among \(f_1(z), \dots, f_s(z)\) which are algebraically independent over the field \(\mathbb{C}(z)\); furthermore, \(\gamma\) and \(C\) are positive constants, and \(\gamma\) depends only on the functions \(f_i(z)\) (i.e. on the algebraic character connected with these functions), and \(C\) only on \(d,\alpha\), and the functions \(f_i(z)\).
11J91 Transcendence theory of other special functions
11J85 Algebraic independence; Gel’fond’s method