Decomposable polynomials in second order linear recurrence sequences

Clemens Fuchs,Dijana Kreso,Christina Karolus

doi:10.1007/s00229-018-1070-8

Clemens Fuchs, Dijana Kreso + Show 1 more

Open Access

https://doi.org/10.1007/s00229-018-1070-8

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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=gcirc h for some g, hin {{mathbb {C}}}[x] satisfying deg g,deg h>1. Under certain assumptions, and provided that h is not of particular type, we show that deg g may be bounded by a constant independent of n, depending only on the sequence.

Highlights

Introduction and resultsLet d ≥ 2 be an integer
We consider a sequence of polynomials (Gn)∞ n=0 in C[x] satisfying the d-th order linear recurrence relation
We focus on decomposable polynomials in second order linear recurrence sequences

Summary

Introduction and results

Consider for example the sequence (Fn(h(x)))∞ n=0, where Fn is defined by (3) and h ∈ C[x] This sequence satisfies a second order linear recurrence relation and we clearly cannot bound deg Fn independently of n. Note that if h is not cyclic and A0(x) = a0 ∈ C, π1π2 = π ∈ C, there exists a vanishing subsum of (4) and one cannot apply the theorem in question; for example, this is the case for Chebyshev polynomials Tn. We state our main result. We remark that likewise, using our results, one may study Diophantine equations of this type where f and/or g are elements of a second order linear recurrence sequence of polynomials. Our proof of Theorem 1 involves applying the theory of S-unit equations over function fields

Polynomial decomposition via Galois theory

Preliminaries and auxiliary results

Proof of Theorem 2

Proof of Theorem 1