Transcendence Criterion for Values of Certain Functions of Several Variables

Let Phi(0)(z) be the function de. ned by Phi(0)(z) = Phi(0)(z(1),...,z(m)) = Sigma(k >= 0) Ek(z(1)(rk),...,z(m)(rk))/F(k)(z(1)(rk),...,z(m)(rk)), where E(k)(z) and F(k)(z) are polynomials in m variables z = (z(1),...,z(m)) with coefficients satisfying a weak growth condition and r >= 2 a fixed integer. For an algebraic point alpha satisfying some conditions, we prove 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 Duverney and Nishioka in one variable case. As applications, we give some examples of transcendental numbers.