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 $.

中文翻译：

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Pillai 问题的函数域变体

在本文中，我们考虑了函数域 $ F $ 上的 Pillai 问题的变体，该问题位于 $ \mathbb{C} $ 上的一个变量中。对于给定的简单线性递推序列$G_n$和$H_m$，定义在$F$上并满足一些弱条件，我们将证明方程$G_n-H_m=f$只有有限多个解$(n,m)\ in \mathbb{N}^2 $ 对于任何非零 $ f \in F $，可以有效地有界。此外，我们证明在适当的假设下，只有有限多个有效可计算的 $ f $ 具有不止一种 $ G_n - H_m $ 形式的表示。