On the Positivity Problem for Simple Linear Recurrence Sequences

TL;DR

This paper proves that determining whether all terms of certain simple linear recurrence sequences are positive is decidable for sequences of order up to 9, with the problem's complexity situated within the Counting Hierarchy.

Contribution

It establishes the decidability of the Positivity Problem for simple LRS of order at most 9, expanding the understanding of the problem's computational boundaries.

Findings

Positivity is decidable for simple LRS of order ≤ 9.

The complexity of the problem lies within the Counting Hierarchy.

The result applies specifically to sequences with no repeated roots in their characteristic polynomial.

Abstract

Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy.