Algebraic function solutions of a certain class of functional equations
The following is a special case of the result proved in the paper. Suppose \(f(z_ 1,...,z_ n)\) in \({\mathbb C}((z_ 1,...,z_ n))\) satisfies the functional equation \[ f(z_ 1^{p_ 1},...,z_ n^{p_ n})=F(z_ 1,...,z_ n,f(z_ 1,...,z_ n)), \] where \(F\) is a rational function. If \(f\) is algebraic over \({\mathbb C}(z_ 1,...,z_ n)\), then it is in fact in \({\mathbb C}(z_ 1,...,z_ n)\). Theorems of this type are of interest in connection with Mahler's method in transcendence theory. The theorem is proved by a nice application of some differential algebra. For background comment and other results of this type obtained by less elegant methods, see the articles by the reviewer and \textit{A. J. van der Poorten}, pp. 211--226 (Zbl 0378.10020) and \textit{K. K. Kubota}, pp. 227-- 229 (Zbl 0378.10021) in ''Transcendence theory: advances and applications'', Proc. Conf., Cambridge 1976 (1977; Zbl 0357.00010).
scientific article; zbMATH DE number 3906608