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.