Decidability of Skolems Problem for Integer Linear Recurrence Sequences

Determine whether there exists an algorithm that, given a linear recurrence sequence over the ring of integers (i.e., an integer linear recurrence sequence specified by integer coefficients and initial values), decides whether there exists an index n \in \mathbb{N} such that u_n = 0.

Background

Linear recurrence sequences (LRS) are sequences defined by a fixed order linear recursion with coefficients in a ring. A central decision problem asks whether an LRS ever attains the value zero, known as Skolems problem.

The paper recalls that, despite partial progress (decidability known for small orders), the general decidability of Skolems problem for LRS over the integers remains unresolved.