A generalization of the method of global relation
Let \(K\) be an algebraic number field, let \(f_i(z)\) be formal power series over \(K\), and take \(\xi\in K\). According to \textit{E. Bombieri} [Recent Progress in Analytic Number Theory, Symp. Durham 1979, Vol. 2, 1--67 (1981; Zbl 0461.10031)], a polynomial relation \(P(f_1(\xi),\dots,f_m(\xi))=0\) over \(K\) is called global if it is true in every completion (at a finite prime) of \(K\), whenever it makes sense in that field. A global relation is called trivial, if it is a consequence of a relation between the power series. In two earlier papers [Math. Notes 48, No. 2, 795--798; translation from Mat. Zametki 48, No. 2, 123--127 (1990; Zbl 0764.11031) and Mosc. Univ. Math. Bull. 44, No. 5, 41--44; translation from Vestn. Mosk. Univ., Ser. I 1989, No. 5, 33--36 (1989; Zbl 0699.10052)], the author investigated \(F\)-series over \(K\). They are of the form \(\sum_{n=0}^\infty a_nn!z^n\), where the absolute values of \(a_n\) and its conjugates do not exceed \(C^n\) and the denominator of \(a_n\) is restricted in a certain way. Here \(C\) is a suitable positive constant. The present paper is a continuation of the earlier ones. The \(F\)-series \(f_i(z)\) are assumed to form a set of solutions of a system of linear differential equations, and \(\xi\) is of the form \(\sum_{k=0}^\infty\theta_k\), where \(\theta_k\in K\). The author proves that under certain complicated conditions, there are no nontrivial global relations between \(f_1(\xi),\dots,f_m(\xi).\)
scientific article; zbMATH DE number 2187248