Transcendence Criterion for Values of Certain Functions of Several Variables

Let ${Phi_0(boldmath{z})}$ be the function defined by $$Phi_0({boldmath z}) = Phi _{0}(z_1,ldots, z_m)=sum_{kgeq 0}frac{E_k(z_1^{r^k},ldots,z_m^{r^k})}{F_k(z_1^{r^k},ldots,z_m^{r^k})},$$ 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.