Algebraic independence of Mahler functions

Abstract.

Suppose that $f_1, \ldots , f_m$ satisfy functional equations of type¶¶$f_i({z^d}) = P_i(z, f_i(z)) \quad {or} \quad f_i(z) = P_i(z, f_i({z^d}))$¶for $i = 1, \ldots , m$, an integer d > 1 and polynomials $P_i \in \Bbb C (z)[ {y}]$ with pairwise distinct partial degrees $\deg _y( {P_1}), \ldots , \deg _y( {P_m})$. Generalizing a result of Keiji Nishioka and using an idea of Kumiko Nishioka we show, that $f_1, \ldots , f_m$ can only be algebraically dependent over $\Bbb C (z)$, if there is an index $\kappa \in \{ {1, \ldots , m}\}$ such that $f_{\kappa }$ is rational.