Skolem Meets Schanuel

TL;DR

This paper introduces an algorithm to solve the Skolem Problem for simple linear recurrence sequences, providing certificates of correctness, and relies on classical number-theoretic conjectures for termination guarantees.

Contribution

It presents a novel algorithm for the Skolem Problem with certificates, assuming classical conjectures, advancing the decidability understanding for simple linear recurrence sequences.

Findings

Algorithm produces correct zero sets with witnesses.

Assumes Skolem and p-adic Schanuel conjectures for termination.

Preliminary experiments show practical applicability.

Abstract

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 \textsc{Skolem} point to the practical applicability of this method.