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On the $p$-adic Skolem Problem

Piotr Bacik, Joël Ouaknine, David Purser, James Worrell

TL;DR

This work studies the $p$-adic zeros of linear recurrence sequences and provides algorithms to decide whether a given non-degenerate LRS has a $p$-adic zero and to compute all such zeros, under the assumption of the $p$-adic Schanuel Conjecture. The authors develop a $p$-adic interpolation framework $f_ullet: Z_p o Z_p$ to extend LRS subsequences and apply Hensel's lemma and Newton polygon techniques to locate and certify zeros, including an important lemma showing that coprime exponential polynomials have only rational common $p$-adic zeros. This leads to decidability and computability results for the $p$-adic Skolem Problem and yields decidability results for the Simultaneous Skolem Problem for coprime LRS, with practical implementation in the Skolem Tool. The methodology also covers algebraic-valued LRS via reduction to the integer case and discusses connections to Skolem-type local-global principles, including the Skolem Conjecture. The work demonstrates the potential of $p$-adic methods to guide zero-detection strategies and to facilitate rational or integer zero discovery under certain hypotheses.

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.

On the $p$-adic Skolem Problem

TL;DR

This work studies the -adic zeros of linear recurrence sequences and provides algorithms to decide whether a given non-degenerate LRS has a -adic zero and to compute all such zeros, under the assumption of the -adic Schanuel Conjecture. The authors develop a -adic interpolation framework to extend LRS subsequences and apply Hensel's lemma and Newton polygon techniques to locate and certify zeros, including an important lemma showing that coprime exponential polynomials have only rational common -adic zeros. This leads to decidability and computability results for the -adic Skolem Problem and yields decidability results for the Simultaneous Skolem Problem for coprime LRS, with practical implementation in the Skolem Tool. The methodology also covers algebraic-valued LRS via reduction to the integer case and discusses connections to Skolem-type local-global principles, including the Skolem Conjecture. The work demonstrates the potential of -adic methods to guide zero-detection strategies and to facilitate rational or integer zero discovery under certain hypotheses.

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 -adic zeros. In this paper we consider the problem of determining whether a given LRS has a -adic zero, as well as the corresponding function problem of computing exact representations of all -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 -adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the -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 -adic Schanuel Conjecture.

Paper Structure

This paper contains 20 sections, 9 theorems, 17 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Skolem Problem
  3. The Bi-Skolem Problem
  4. The Rational Skolem Problem
  5. The p-adic Skolem Problem
  6. Main Results
  7. Preliminaries
  8. p-adic numbers
  9. Computing with p-adic numbers
  10. p-adic interpolation of linear recurrence sequences
  11. Decidability of the p-adic Skolem Problem
  12. Informal Outline of the Algorithm
  13. Simultaneous Zeros of Coprime Exponential Polynomials
  14. The Algorithm and the Proof of Termination and Correctness
  15. Simultaneous Skolem Problem
  16. ...and 5 more sections

Key Result

Theorem 1

Assuming the $p$-adic Schanuel Conjecture, $\mathtt{SP}(\mathbb{Z}_p)$ is decidable and $\mathtt{FSP}(\mathbb{Z}_p)$ is computable.

Figures (1)

  • Figure 1: (a)-(c): Timing comparison between the Hensel-only and full algorithm (in seconds) in cases where both succeed within 60s. (d): Distribution of instances requiring a prime in a given range, and the instances for which the Hensel-only algorithm terminates in 60s (displayed up to 900); the system is highly effective up to primes of around 400-500 in this time limit.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Hensel's Lemma for power series
  • Definition 4
  • Theorem 5
  • Conjecture 6: The $p$-adic Schanuel Conjecture
  • Lemma 6
  • Remark 7
  • Remark 8
  • Remark 9
  • ...and 10 more