Abstract
AbstractFor $R(z, w)\in \mathbb {C}(z, w)$ 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.
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