On the Skolem Problem for Reversible Sequences

TL;DR

This paper provides an alternative proof that the Skolem Problem is decidable for reversible sequences of order up to seven, using advanced Galois conjugate analysis on concentric circles.

Contribution

It introduces a new proof technique for the decidability of the Skolem Problem in a specific class of sequences, expanding the theoretical understanding.

Findings

Decidability established for reversible sequences of order at most seven.

Utilizes Galois conjugates on concentric circles for the proof.

Provides an alternative approach to previous results.

Abstract

Given an integer linear recurrence sequence $\langle X_n \rangle_n$, 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.