Positivity Problems for Low-Order Linear Recurrence Sequences

TL;DR

This paper proves the decidability of positivity problems for low-order linear recurrence sequences (up to order 5), with complexity results and implications for higher orders related to deep number theory challenges.

Contribution

It establishes decidability results for positivity and ultimate positivity of LRS of order 5 or less, and highlights the complexity and number theory implications of extending these results.

Findings

Decidability of Positivity for LRS of order ≤ 5 in the Counting Hierarchy.

Decidability of Ultimate Positivity for LRS of order ≤ 5 in polynomial time.

Hardness results indicating that extending to order 6 involves major breakthroughs in number theory.

Abstract

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem} (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers.