A function field variant of Pillai's problem

TL;DR

This paper investigates a function field version of Pillai's problem, proving finiteness results for solutions to certain linear recurrence equations over complex function fields, with effective bounds and computability aspects.

Contribution

It establishes finiteness and effective bounds for solutions of linear recurrence difference equations over function fields, extending classical Pillai's problem to this setting.

Findings

Finitely many solutions for G_n - H_m = f over complex function fields.

Effective bounds on the number of solutions.

Finiteness of multiple representations of the same difference.

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 $.