Completing the picture for the Skolem Problem on order-4 linear recurrence sequences

TL;DR

This paper proves that the Skolem Problem, which determines if a linear recurrence sequence has a zero, is decidable for all algebraic sequences of order up to 4, closing a long-standing open question.

Contribution

We establish the decidability of the Skolem Problem for all algebraic linear recurrence sequences of order at most 4, extending previous results to the full order-4 case.

Findings

Decidability of the Skolem Problem for algebraic LRS of order 4.

Completes the classification of decidability for low-order algebraic LRS.

Provides a definitive answer to a century-old open problem.

Abstract

For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem Problem is indeed decidable for algebraic LRS of order at most 3, and real algebraic LRS of order at most 4. However, for general algebraic LRS of order 4 the question of decidability has remained open. Our main contribution in this paper is to prove decidability for this last case, i.e. we show that the Skolem Problem is decidable for all algebraic LRS of order at most 4.