Transcendence Criterion for Values of Certain Functions of Several Variables

Abstract

Let ${\Phi_0(\boldmath{z})}$ be the function defined by

$$\Phi_0({\boldmath z}) = \Phi _{0}(z_1,\ldots, z_m)=\sum_{k\geq 0}\frac{E_k(z_1^{r^k},\ldots,z_m^{r^k})}{F_k(z_1^{r^k},\ldots,z_m^{r^k})},$$

where ${E_k(\boldmath{z})}$ and ${F_k(\boldmath{z})}$ are polynomials in m variables ${\boldmath{z} = (z_1,\ldots, z_m)}$ with coefficients satisfying a weak growth condition and r ≥ 2 a fixed integer. For an algebraic point ${\boldmath{\alpha}}$ satisfying some conditions, we prove that ${\Phi_{0}(\boldmath{\alpha})}$ is algebraic if and only if ${\Phi_{0}(\boldmath{z})}$ is a rational function. This is a generalization of the transcendence criterion of Duverney and Nishioka in one variable case. As applications, we give some examples of transcendental numbers.