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Skolem Meets Schanuel

Authors Yuri Bilu , Florian Luca , Joris Nieuwveld, Joël Ouaknine , David Purser , James Worrell

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Author Details

Yuri Bilu
  • Institut de Mathématiques de Bordeaux, Université de Bordeaux and CNRS, Talence, France
Florian Luca
  • School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
  • Research Group in Algebraic Structures & Applications, King Abdulaziz University, Saudi Arabia
  • Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
Joris Nieuwveld
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
David Purser
  • University of Warsaw, Poland
James Worrell
  • Department of Computer Science, University of Oxford, UK

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Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell. Skolem Meets Schanuel. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.

Subject Classification

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


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