$ S $-unit values of $ G_n + G_m $ in function fields

TL;DR

This paper establishes bounds on indices for sums of terms in a linear recurrence over function fields that are $S$-units, extending known number field results to the function field setting.

Contribution

It provides the first upper bounds for indices in linear recurrences over function fields where the sum is an $S$-unit, paralleling number field results.

Findings

Upper bounds on indices n and m for G_n + G_m as S-units

Extension of number field results to function fields

Demonstrates analogous behavior in function fields

Abstract

In this paper we consider a simple linear recurrence sequence $ G_n $ defined over a function field in one variable over the field of complex numbers. We prove an upper bound on the indices $ n $ and $ m $ such that $ G_n + G_m $ is an $ S $-unit. This is a function field analogue of already known results in number fields.