On the size of a linear combination of two linear recurrence sequences over function fields

TL;DR

This paper investigates bounds on the valuation of differences of two non-degenerate linear recurrence sequences over function fields, establishing effective bounds under certain conditions.

Contribution

It provides the first effective bounds on the valuation of the difference of two linear recurrence sequences over function fields.

Findings

Established bounds for valuations of sequence differences

Identified conditions for effective computability of bounds

Extended results to sequences over complex function fields

Abstract

Let $ G_n $ and $ H_m $ be two non-degenerate linear recurrence sequences defined over a function field $ F $ in one variable over $ \mathbb{C} $, and let $ \mu $ be a valuation on $ F $. We prove that under suitable conditions there are effectively computable constants $ c_1 $ and $ C' $ such that the bound \begin{equation*} \mu(G_n - H_m) \leq \mu(G_n) + C' \end{equation*} holds for $ \max \{n,m\} > c_1 $.