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Skolem Meets Bateman-Horn

Florian Luca, James Maynard, Armand Noubissie, Joël Ouaknine, James Worrell

TL;DR

The paper addresses the Skolem Problem for integer linear recurrence sequences by introducing Universal Skolem Sets, sets of indices for which zero-terms can be decided effectively. It constructs such a set ${\mathcal S}$ with unconditional lower density at least $0.29$ and density one under the Bateman--Horn conjecture, linking decidability to deep prime-distribution heuristics. The main technical strategy fuses explicit bounds on polynomial–exponential Diophantine equations with primality-pattern results, leveraging tools from height theory, multiplicative-group equations, and Bateman--Horn-type conjectures. This approach opens a new avenue for approaching Skolem decidability and highlights surprising connections between LRS zeros, Diophantine geometry, and prime-value distribution of linear forms.

Abstract

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and control theory, amongst many others. Decidability of the Skolem Problem is notoriously open. The state of the art is a decision procedure for recurrences of order at most 4: an advance achieved some 40 years ago, based on Baker's theorem on linear forms in logarithms of algebraic numbers. A new approach to the Skolem Problem was recently initiated via the notion of a Universal Skolem Set: a set $S$ of positive integers such that it is decidable whether a given non-degenerate linear recurrence sequence has a zero in $S$. Clearly, proving decidability of the Skolem Problem is equivalent to showing that $\mathbb{N}$ itself is a Universal Skolem Set. The main contribution of the present paper is to construct a Universal Skolem Set that has lower density at least $0.29$. We show moreover that this set has density one subject to the Bateman-Horn conjecture. The latter is a central unifying hypothesis concerning the frequency of prime numbers among the values of systems of polynomials.

Skolem Meets Bateman-Horn

TL;DR

The paper addresses the Skolem Problem for integer linear recurrence sequences by introducing Universal Skolem Sets, sets of indices for which zero-terms can be decided effectively. It constructs such a set with unconditional lower density at least and density one under the Bateman--Horn conjecture, linking decidability to deep prime-distribution heuristics. The main technical strategy fuses explicit bounds on polynomial–exponential Diophantine equations with primality-pattern results, leveraging tools from height theory, multiplicative-group equations, and Bateman--Horn-type conjectures. This approach opens a new avenue for approaching Skolem decidability and highlights surprising connections between LRS zeros, Diophantine geometry, and prime-value distribution of linear forms.

Abstract

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and control theory, amongst many others. Decidability of the Skolem Problem is notoriously open. The state of the art is a decision procedure for recurrences of order at most 4: an advance achieved some 40 years ago, based on Baker's theorem on linear forms in logarithms of algebraic numbers. A new approach to the Skolem Problem was recently initiated via the notion of a Universal Skolem Set: a set of positive integers such that it is decidable whether a given non-degenerate linear recurrence sequence has a zero in . Clearly, proving decidability of the Skolem Problem is equivalent to showing that itself is a Universal Skolem Set. The main contribution of the present paper is to construct a Universal Skolem Set that has lower density at least . We show moreover that this set has density one subject to the Bateman-Horn conjecture. The latter is a central unifying hypothesis concerning the frequency of prime numbers among the values of systems of polynomials.
Paper Structure (15 sections, 9 theorems, 83 equations)

This paper contains 15 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Number fields
  4. Heights
  5. Linear equations in elements of multiplicative groups
  6. Mean Value of Multiplicative Functions
  7. Distribution of primes
  8. A Universal Skolem Set
  9. The Density of ${\mathcal{S}}$
  10. Counting correlated representations
  11. Counting all representations
  12. Deferred Proofs
  13. Proof of Claim \ref{['claim:vanishing']}
  14. Proof of Claim \ref{['claim:heightbound']}
  15. Proof of Claim \ref{['claim:easy']}

Key Result

Theorem 1

Let $\alpha,\beta,\alpha_1,\ldots,\alpha_s$ be non-zero algebraic numbers for some $s \geq 2$. Then

Theorems & Definitions (21)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture 6: Bateman-Horn
  • Theorem 7
  • Claim 7
  • Claim 7
  • Claim 7
  • ...and 11 more