Skolem Meets Bateman-Horn

TL;DR

This paper introduces a new approach to the Skolem Problem by constructing a Universal Skolem Set with significant lower density, and explores its density properties under the Bateman-Horn conjecture, advancing understanding of decidability in linear recurrence sequences.

Contribution

The paper constructs a Universal Skolem Set with density at least 0.29 and shows it has density one assuming the Bateman-Horn conjecture, providing new insights into the decidability of the Skolem Problem.

Findings

Constructed a Universal Skolem Set with density ≥ 0.29.

Proved the set has density one under Bateman-Horn conjecture.

Advances the approach to the Skolem Problem via density and number theory.

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.