Positivity Proofs for Linear Recurrences through Contracted Cones

TL;DR

This paper introduces a novel method using contracted cones and an extension of Perron-Frobenius theory to decide positivity of linear recurrence sequences with polynomial coefficients, applicable to a broad class of recurrences.

Contribution

It develops an algorithm that constructs contracted cones to prove positivity of linear recurrences with polynomial coefficients, extending classical Perron-Frobenius theory.

Findings

Algorithm successfully decides positivity outside a hyperplane.

Constructs cones invariant under recurrence operators.

Extends Perron-Frobenius theory to polynomial coefficient matrices.

Abstract

Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to~5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.