## Algebraic independence of Mahler functions and their values.(English)Zbl 0852.11036

The paper deals with the algebraic independence of a special class of Mahler functions in several complex variables, namely functions $$f(z)= (f_1 (z), \dots, f_m (z))$$ satisfying a functional equation $$f(z)= Af(\Omega z)+ b(z)$$, where $$A$$ is a constant matrix, the entries of $$b(z)$$ are rational functions and everything is defined over a number field $$K$$. The transformation $$\Omega$$ on $$z= (z_1, \dots, z_n)$$ is defined by $$(\Omega z)_i= \prod^n_{j=1} z_j^{\omega_{ij}}$$. Assume that $$\alpha$$ is an algebraic point, $$\Omega^k \alpha\to 0$$ as $$k\to \infty$$ at the correct rate and $$f(\Omega^k \alpha) \neq 0$$ for infinitely many $$k$$. Then
(1) If $$f_1, \dots, f_m$$ are algebraically dependent over $$\mathbb{C}(z_1, \dots, z_n)$$ there are constants $$c_1, \dots, c_m$$ such that $$c_1 f_1+ \dots+ c_m f_m$$ is in $$\mathbb{C}(z_1, \dots, z_n)$$.
(2) If $$f_1, \dots, f_m$$ are algebraically independent then so are $$f_1 (\alpha), \dots, f_m (\alpha)$$.
The arguments are similar to those in the original papers of Mahler. As an interesting example in one variable, let $$g(z)= \sum^\infty_{k=0} \alpha^{d^k} x^k$$ where $$d$$ is an integer greater than 1 and $$\alpha$$ is a non-zero algebraic number with absolute value less than 1. Then the numbers $$g^{(l)} (\beta)$$ for $$l= 0, 1, 2, \dots$$ and $$\beta$$ non-zero algebraic are algebraically independent. Mahler’s example $$F_\omega (z_1, z_2)= \sum^\infty_{h_1= 1} \sum_{h_2= 1}^{[ h_1 \omega]} z_1^{h_1} z_2^{h_2}$$ leads similarly to the algebraic independence of the numbers $${{\partial^{l_1+ l_2}} \over {\partial z_1^{l_1} \partial z_2^{l_2}}} F_\omega (\alpha_1, \alpha_2)$$.

### MSC:

 11J85 Algebraic independence; Gel’fond’s method 39B32 Functional equations for complex functions
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### References:

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