For $R(z, w)\in \mathbb {C}(z, w)$ of degree at least 2 in w, we show that the number of rational functions $f(z)\in \mathbb {C}(z)$ solving the difference equation $f(z+1)=R(z, f(z))$ is finite and bounded just in terms of the degrees of R in the two variables. This complements a result of Yanagihara, who showed that any finite-order meromorphic solution to this sort of difference equation must be a rational function. We prove a similar result for the differential equation $f'(z)=R(z, f(z))$ , building on a result of Eremenko.

中文翻译：

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某些微分和差分方程解的有效有限性

对于w中至少 2 次的 $R(z, w)\in \mathbb {C}(z, w)$ ，我们证明有理函数 $f(z)\in \mathbb {C}( z)$ 求解差分方程 $f(z+1)=R(z, f(z))$ 是有限的，并且仅根据两个变量中R的度数有界。这补充了 Yanagihara 的结果，他表明这种差分方程的任何有限阶亚纯解都必须是有理函数。基于 Eremenko 的结果，我们证明了微分方程 $f'(z)=R(z, f(z))$ 的类似结果。