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

TL;DR

This paper establishes an upper bound on the degree of polynomial identities involving linear recurrence sequences, extending previous results and showing the bound depends only on sequence parameters, not on the polynomial degree.

Contribution

The paper generalizes prior work by proving a bound on polynomial degrees in identities involving multiple linear recurrence sequences, independent of the polynomial's degree.

Findings

Degree of polynomial R(z) is bounded by an effectively computable constant.

The bound depends only on sequence parameters and coefficient bounds, not on the polynomial degree.

Generalizes previous results to include multiple recurrence sequences and polynomial transformations.

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), $R(z) \in \mathbb{C}[z]$, 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 $k$).