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Effective finiteness of solutions to certain differential and difference equations (2011.02968v2)
Published 5 Nov 2020 in math.NT and math.CV
Abstract: For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(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.