On an upper bound of the degree of polynomial identities regarding linear recurrence sequences

Abstract

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several interesting identities involving this sequence such as $F_n^2+F_{n+1}^2=F_{2n+1}$, for all $n\geq 0$. Inspired by this naive identity, in 2012, Chaves, Marques and Togb\'e proved that if $(G_m)_m$ is a linear recurrence sequence (under weak assumptions) and $G_n^s+\cdots +G_{n+k}^s\in (G_m)_m$, for infinitely many positive integers $n$, then $s$ is bounded by an effectively computable constant depending only on $k$ and the parameters of $G_m$. In this paper, we generalize this result, proving, in particular, that if $(G_m)_m$ and $(H_m)_m$ are linear recurrence sequences (also under weak assumptions), $R(z) \in \mathbb{C}[z]$ is a monic polynomial, and $\epsilon_0R(G_n)+\epsilon_1R(G_{n+1})+\cdots +\epsilon_{k-1}R(G_{n+k-1})+R(G_{n+k})$ belongs to $(H_m)_m$, for infinitely many positive integers $n$, then the degree of $R(z)$ is bounded by an effectively computable constant depending only on the upper and lower bounds of the $\epsilon_i$'s and the parameters of $G_m$ (but surprisingly not on $k$)