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arXiv · 2408.09448

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

Darsana N·S. S. Rout
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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.

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Darsana N, S. S. Rout. 2024-08-18. Zeros and $S$-units in sums of terms of recurrence sequences in function fields. https://arxiv.org/abs/2408.09448

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