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$f_1, \ldots , f_m$\end{document} satisfy functional equations of type¶¶\documentclass[12pt]{minimal}
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$f_i({z^d}) = P_i(z, f_i(z)) \quad {or} \quad f_i(z) = P_i(z, f_i({z^d})) $\end{document}¶for \documentclass[12pt]{minimal}
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$i = 1, \ldots , m$\end{document}, an integer d > 1 and polynomials \documentclass[12pt]{minimal}
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$P_i \in \Bbb C (z)[ {y}]$\end{document} with pairwise distinct partial degrees \documentclass[12pt]{minimal}
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$\deg _y( {P_1}), \ldots , \deg _y( {P_m})$\end{document}. Generalizing a result of Keiji Nishioka and using an idea of Kumiko Nishioka we show, that \documentclass[12pt]{minimal}
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$f_1, \ldots , f_m$\end{document} can only be algebraically dependent over \documentclass[12pt]{minimal}
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$\Bbb C (z)$\end{document}, if there is an index \documentclass[12pt]{minimal}
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$\kappa \in \{ {1, \ldots , m}\}$\end{document} such that \documentclass[12pt]{minimal}
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$f_{\kappa }$\end{document} is rational.
