On the Skolem Problem for Reversible Sequences

Abstract

Given an integer linear recurrence sequence $⟨X_n{⟩}_n$, the Skolem Problem asks to determine whether there is a natural number $n$ such that $X_n=0$. Recent work by Lipton, Luca, Nieuwveld, Ouaknine, Purser, and Worrell proved that the Skolem Problem is decidable for a class of reversible sequences of order at most seven. Here we give an alternative proof of their result. Our novel approach employs a powerful result for Galois conjugates that lie on two concentric circles due to Dubickas and Smyth.

arXiv comment15 pages, accepted for publication at MFCS 2022