Perfect powers in polynomial power sums

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.