Perfect powers in polynomial power sums

TL;DR

This paper proves that non-degenerate simple polynomial linear recurrence sequences of order at least two cannot contain arbitrarily large powers of polynomials, and in the binary case, only finitely many sequence elements are perfect powers.

Contribution

It establishes bounds on the exponents and indices of polynomial powers within such recurrence sequences, extending understanding of polynomial perfect powers in these sequences.

Findings

No large polynomial powers appear in sequences of order ≥ 2.

In binary sequences, only finitely many elements are perfect powers.

The bounds depend solely on the sequence's parameters.

Abstract

We prove that a non-degenerate simple linear recurrence sequence $ (G_n(x))_{n=0}^{\infty} $ of polynomials satisfying some further conditions cannot contain arbitrary large powers of polynomials if the order of the sequence is at least two. In other words we will show that for $ m $ large enough there is no polynomial $ h(x) $ of degree $ \geq 2 $ such that $ (h(x))^m $ is an element of $ (G_n(x))_{n=0}^{\infty} $. The bound for $ m $ depends here only on the sequence $ (G_n(x))_{n=0}^{\infty} $. In the binary case we prove even more. We show that then there is a bound $ C $ on the index $ n $ of the sequence $ (G_n(x))_{n=0}^{\infty} $ such that only elements with index $ n \leq C $ can be a proper power.