Remarks on a paper by U. Zannier
The author considers some problems related to the results obtained by \textit{U. Zannier} [Aequationes Math. 55, No. 1-2, 15-43 (1998; Zbl 0894.39007)] for a Laurent series satisfying a functional equation of the form \[ f(x^m)= P\bigl(x,f(x) \bigr), \tag{1} \] where \(P(x,y)\in \mathbb{C}(x)[y]\). Zannier proved that if \(f(x)\) is not a rational function, then the set of positive integers \(m\) such that a relation (1) holds for some \(P=P_m\) consists precisely of the powers of a single integer. The author considers the case \[ f(x)= Q\bigl(x,f(x^m) \bigr). \tag{2} \] Assume that \(f(x)\) is a formal Laurent series with complex coefficients satisfying the functional equation (2) for an integer \(m\geq 2\) and \(Q(x,y)\in \mathbb{C}(x)[y]\). Let us denote by \(B_f\) the set of positive integers \(m\) such that the equation (2) holds true for some \(Q=Q_m\). Then the following result is proved. Theorem 1. Let \(f(x)\) be a Laurent series with complex coefficient, not representing a rational function, and let \(a,b\in B_f\). Then \(a,b\) are multiplicatively dependent. The paper contains also some generalization of \textit{K. Nishioka}'s theorem for Polya's functional equation [Arch. Math. 44, 330-335 (1985; Zbl 0568.12014)]. This result reads as follows. Theorem 2. Suppose that \(f(x)\in \mathbb{C} ([x])\) satisfies the functional equation \[ f(x^m)= \varphi\bigl[ x,f(x), \dots,f(x^{m-1}) \bigr], \tag{3} \] where \(\varphi(x,y_1, \dots,y_{m-1})\in \mathbb{C}(x,y_1, \dots, y_{m-1})\). If \(f(x)\) is algebraic over \(\mathbb{C}(x)\), then \(f(x)\in \mathbb{C}(x)\). Moreover, interesting examples of equations of type (3) are presented in the paper.
scientific article; zbMATH DE number 1538473