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arXiv · 2008.10339

A function field variant of Pillai's problem

Clemens Fuchs·Sebastian Heintze
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Abstract

In this paper, we consider a variant of Pillai's problem over function fields $ F $ in one variable over $ \mathbb{C} $. For given simple linear recurrence sequences $ G_n $ and $ H_m $, defined over $ F $ and satisfying some weak conditions, we will prove that the equation $ G_n - H_m = f $ has only finitely many solutions $ (n,m) \in \mathbb{N}^2 $ for any non-zero $ f \in F $, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable $ f $ with more than one representation of the form $ G_n - H_m $.

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Clemens Fuchs, Sebastian Heintze. 2020-08-24. A function field variant of Pillai's problem. https://doi.org/10.1016/j.jnt.2020.11.004

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