Effective finiteness of solutions to certain differential and difference equations
Number Theory
2021-01-25 v2 Complex Variables
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.
Cite
@article{arxiv.2011.02968,
title = {Effective finiteness of solutions to certain differential and difference equations},
author = {Patrick Ingram},
journal= {arXiv preprint arXiv:2011.02968},
year = {2021}
}
Comments
Minor corrections made based on referee feedback, and a few remarks/examples/questions added