Transcendence criterion for values of certain functions of several variables
Let \(\Phi_0(z)\) be the function defined by \[ \Phi_0(z)= \Phi_0(z_1,\dots, z_m)= \sum_{k\geq 0} {E_k(z^{r_k}_1,\dots, z^{r^k}_m)\over F_k(z^{r_k}_1,\dots, z^{r_k}_m)}, \] where \(E_k(z)\) and \(F_k(z)\) are polynomials in \(m\) variables \(z= (z_1,\dots, z_m)\) with coefficients satisfying a weak growth condition and \(r\geq 2\) a fixed integer. For an algebraic point \(\alpha\) satisfying some conditions, the author proves that \(\Phi_0(\alpha)\) is algebraic if and only if \(\Phi_0(z)\) is a rational function. This is a generalization of the transcendence criterion of \textit{D. Duverney} and \textit{K. Nishioka} [Acta Arith. 110, No. 4, 305--330 (2003; Zbl 1049.11078)]. As applications, some new examples of transcendental numbers are obtained. For example, the number \[ \sideset\and {'}\to\sum_{k\geq 0}a_k/(2^{r^k}+ 3^{r^k}+ b_k) \] is transcendental, where \(a_k,b_k\in\mathbb K\) (an algebraic number field), \(\log\| a_k\|,\log\| b_k\|= o(r^k)\), and \(a_k,2^{r^k}+ 3^{r^k}+ b_k\neq 0\) for infinitely many \(,\) and \(\sideset\and {'}\to\sum\) means taking the sum over these \(k\).
scientific article; zbMATH DE number 5703366
scientific article