Algebraic independence of Mahler functions and their values (Q1915241)

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 \({{\partial^{l_1+ l_2}} \over {\partial z_1^{l_1} \partial z_2^{l_2}}} F_\omega (\alpha_1, \alpha_2)\).