A New Method for the Investigation of Arithmetical Properties of Analytic Functions

Abstract

The general scheme of the proof uses the action of the Galois group and a random walk. We consider auxiliary functions of the form Fj(z) = P(z + Xi; f(z)) for algebraic xi such that each number conjugate to Fk>)(w) for w e S has the same form Fk)(w). Then we evaluate the order ui of the zero of Fj(z) at z = w and show that uj satisfies a system of inequalities (3.14) (or (W)). This system of inequalities describes a random walk in a multidimensional cube (discrete Markov chain), and using generating functions, we show that IS I ? p. Section 1 consists of a number of auxiliary lemmas (cf. [3]-[5]). In Section 2 we construct the numbers Xj using the group ring Z[G] and in Section 3 we investigate the auxiliary functions Fj(z) and establish connections between the u4. Finally, in Section 4 it is proved that the random