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

Abstract

Let $(U_n{)}_{n\ge 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and ${\mathcal{O}}_S^{\ast }$ 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 $U_{n_1}+\cdots +U_{n_r}\in {\mathcal{O}}_S^{\ast }$ in nonnegative integers $n_1,\dots ,n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_{\ell }=0$ in $(n,m,\ell )\in {\mathbb{N}}^3$, where $U_n,V_m,W_{\ell }$ are simple linear recurrence sequences in the function field.

19 pages