# Decomposable polynomials in second order linear recurrence sequences

## TL;DR

This paper investigates when elements of second order linear recurrence polynomial sequences can be decomposed into compositions of lower-degree polynomials, establishing bounds on the degrees of such decompositions under specific conditions.

## Contribution

It provides new bounds on the degrees of decomposable polynomials within second order linear recurrence sequences, under certain assumptions and excluding special cases.

## Key findings

- Bound on degree of g independent of n
- Conditions under which decomposition is limited
- Exclusion of particular types of h

## Abstract

We study elements of second order linear recurrence sequences $(G_n)_{n= 0}^{\infty}$ of polynomials in $\mathbb{C}[x]$ which are decomposable, i.e. representable as $G_n=g\circ h$ for some $g, h\in \mathbb{C}[x]$ satisfying $\operatorname{deg}g,\operatorname{deg}h>1$. Under certain assumptions, and provided that $h$ is not of particular type, we show that $\operatorname{deg}g$ may be bounded by a constant independent of $n$, depending only on the sequence.