Effective Divergence Analysis for Linear Recurrence Sequences

TL;DR

This paper investigates the growth and divergence properties of rational linear recurrence sequences, providing polynomial-time algorithms for deciding divergence and computing precise growth bounds for low-order sequences.

Contribution

It introduces efficient algorithms for divergence decision and growth rate estimation specifically for low-order rational linear recurrence sequences.

Findings

Divergence is decidable in polynomial time for low-order sequences.

A polynomial-time algorithm computes effective lower bounds on growth rates.

The methods apply to rational linear recurrence sequences, enhancing understanding of their long-term behavior.

Abstract

We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational linear recurrence sequence and computes effective fine-grained lower bounds on the growth rate of the sequence.