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Effective finiteness of solutions to certain differential and difference equations

Part of: Difference equations Qualitative theory Difference and functional equations Arithmetic algebraic geometry

Published online by Cambridge University Press:  16 February 2021

Patrick Ingram
Patrick Ingram*
Affiliation:
York University, Toronto, Canada
*
e-mail: [email protected]
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Abstract

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.

Keywords

Difference equationsdifferential equationsrational functionsarithmetic geometry

MSC classification

Primary: 39A22: Growth, boundedness, comparison of solutions
Secondary: 34C11: Growth, boundedness 11G50: Heights
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 52 - 67
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Copyright
© Canadian Mathematical Society 2021

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