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On the p-adic Skolem Problem

Authors Piotr Bacik , Joël Ouaknine , David Purser , James Worrell

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ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Skolem Problem
  • p-adic Schanuel Conjecture
  • Skolem Conjecture
  • Exponential Local-Global Principle
  • exponential polynomial

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Abstract

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many p-adic zeros. In this paper we consider the problem of determining whether a given LRS has a p-adic zero, as well as the corresponding function problem of computing exact representations of all p-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the p-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the p-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the p-adic Schanuel Conjecture.

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Piotr Bacik, Joël Ouaknine, David Purser, and James Worrell. On the p-adic Skolem Problem. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.8

Author Details

Piotr Bacik
  • University of Oxford, UK
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
David Purser
  • University of Liverpool, UK
James Worrell
  • University of Oxford, UK

Funding

  • Bacik, Piotr: Supported by EPSRC grant EP/X033813/1.
  • Ouaknine, Joël: Supported by ERC grant DynAMiCs (101167561) and DFG grant 389792660 as part of https://perspicuous-computing.science.
  • Worrell, James: Supported by EPSRC grant EP/X033813/1.

Acknowledgements

The authors would like to thank Naoki Fujita for helpful comments on the main algorithm. Joël Ouaknine is also affiliated with Keble College, Oxford as an https://emmy.network Fellow.

Supplementary Materials

References

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