Decidability of the Skolem problem

Establish the decidability of the Skolem problem: given a linear recurrence sequence over the integers (or rationals), determine whether there exists an index at which the sequence equals zero. This problem underpins the full computation of invariant ideals for polynomial loops, which has been shown to be at least as hard as the Skolem problem.

Background

The paper studies computing polynomial loop invariants and notes that fully generating the invariant ideal is at least as hard as the Skolem problem. Since the Skolem problem asks whether a given linear recurrence sequence ever takes the value zero, its unresolved status directly impacts the feasibility of deciding invariant ideals in general.

Because the decidability of the Skolem problem has remained open for nearly a century, the authors focus on partial computation of invariants within finite-dimensional subspaces, developing algorithms that avoid the intractability suggested by the Skolem link.

References

However, as recently shown in , this task is at least as hard as the Sk\"olem problem, whose decidability has remained widely open for almost a century.

— Algebraic Tools for Computing Polynomial Loop Invariants (Extended Version)  (2412.14043 - Bayarmagnai et al., 2024) in Introduction (Section 1)

Over a field of characteristic $0$, decidability of the Skolem Problem is a longstanding open problem.

— Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization  (2609.03127 - Dong et al., 2 Sep 2026) in Section 1, subsection “The Skolem Problem and a multi-dimensional generalization”

while deciding whether the orbit of a specified rational point ever meets a rational linear hyperplane is equivalent to the classical Skolem problem and remains open.

— An undecidable problem in dynamics  (2610.22123 - Bell, 15 Oct 2026) in Introduction

There are known instances of order-five LRS where is Skolem problem is unresolved, and instances of LRS with three dominant roots whose sign description is provably not almost-periodic [1, Sec. 4], and instances of LRS with three dominant roots for which the positivity and ultimate positivity problems (respectively, does the sign description contain −1, does the sign description contain −1 only finitely many times) are unresolved [12, Sec. 5].

— Model Checking for Noisy Toric Words  (2610.23463 - Vahanwala, 16 Oct 2026) in Section 3, page XX:9