Zeros and $S$-units in sums of terms of recurrence sequences in function fields

TL;DR

This paper investigates solutions to Diophantine equations involving sums of recurrence sequence terms and $S$-units over function fields, establishing finiteness results using advanced Diophantine techniques.

Contribution

It provides effective finiteness results for solutions of specific Diophantine equations involving recurrence sequences and $S$-units in function fields, extending previous work.

Findings

Finiteness of solutions for sums of recurrence terms in $S$-units

Finiteness of solutions for the equation $U_n+V_m+W_ =0$ in recurrence sequences

Application of Brownawell and Masser's results to function fields

Abstract

Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_\ell = 0$ in $(n, m, \ell)\in \N^3$, where $U_n,V_m,W_\ell$ are simple linear recurrence sequences in the function field.