Effective finiteness of solutions to certain differential and difference equations

TL;DR

This paper proves that the number of rational solutions to certain differential and difference equations with rational functions is finite and bounded based on the degrees of the functions involved, extending previous results.

Contribution

It establishes the finiteness and explicit bounds for rational solutions to specific differential and difference equations, complementing prior work on meromorphic solutions.

Findings

Number of rational solutions is finite and bounded by degrees of R

Extends results to differential equations of similar form

Builds on and complements previous theorems by Yanagihara and Eremenko

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.