Algebraic independence of Mahler functions (Q1587791)

The Mahler functions of this paper satisfy functional equations of the type \(f(z^d)= P(z,f(z))\) or \(f(z)= P(z,f(z^d))\) (named after Mahler who first investigated the transcendence properties of functions of this type at algebraic points in the 1930s). Here \(z\) is a single complex variable, \(d>1\) is an integer and \(P\) is a rational function. \textit{K. Nishioka} [Arch. Math. 44, 330-335 (1985; Zbl 0568.12014)] proved that if such a function \(f\) is analytic in a neighbourhood of the origin and algebraic over \({\mathcal C} (z)\), then \(f\) is rational. The main result of this paper is that if \(f_1, \dots, f_m\) are Mahler functions analytic in a neighbourhood of the origin and their degrees (i.e., the maximum of the degree of the numerator and the denominator) are distinct and if \(f_1, \dots, f_m\) are algebraically dependent, then some \(f_k\) is rational. Nishioka's theorem provides the basis for an induction argument to obtain the result.