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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 $$f_i(z)$$.
##### MSC:
 11J91 Transcendence theory of other special functions 11J85 Algebraic independence; Gel’fond’s method
##### Keywords:
E-functions; algebraic dependence; polynomials