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On the Skolem Problem for Reversible Sequences

Author George Kenison

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  • 15 pages

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George Kenison
  • Institute of Logic and Computation, TU Wien, Vienna, Austria


I am grateful to both Adam Keilthy and Florian Luca for their feedback and many helpful discussions. I also thank the anonymous reviewers for their constructive feedback.

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George Kenison. On the Skolem Problem for Reversible Sequences. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 61:1-61:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Given an integer linear recurrence sequence ⟨X_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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Computing methodologies → Algebraic algorithms
  • The Skolem Problem
  • Linear Recurrences
  • Verification


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