Positivity of Nearly Linearly Recurrent Sequences

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Bibliographic Details
Main Authors:	Pouly, Amaury, Shirmohammadi, Mahsa, Worrell, James
Format:	Preprint
Published:	2025
Subjects:	Dynamical Systems Logic in Computer Science F.4.3
Online Access:	Acceder al recurso
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author	Pouly, Amaury Shirmohammadi, Mahsa Worrell, James
author_facet	Pouly, Amaury Shirmohammadi, Mahsa Worrell, James
contents	Nearly linear recurrences are a generalisation of linear recurrences and are instances of linear time-invariant systems in control theory and linear constraint loops in program analysis. In this paper we formulate the Positivity Problem for such recurrences. This asks whether all sequences satisfying a given recurrence with given initial conditions are positive. This problem is a generalisation of the Positivity Problem for linear recurrence sequences, and is a special case of the non-reachability problem for linear time-invariant systems. Our main contribution is a decision procedure for the Positivity Problem for recurrences of order two. The termination proof of our procedure relies on a new transcendence result for infinite series that is of independent interest.
format	Preprint
id	arxiv_https___arxiv_org_abs_2508_00944
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	Positivity of Nearly Linearly Recurrent Sequences Pouly, Amaury Shirmohammadi, Mahsa Worrell, James Dynamical Systems Logic in Computer Science F.4.3 Nearly linear recurrences are a generalisation of linear recurrences and are instances of linear time-invariant systems in control theory and linear constraint loops in program analysis. In this paper we formulate the Positivity Problem for such recurrences. This asks whether all sequences satisfying a given recurrence with given initial conditions are positive. This problem is a generalisation of the Positivity Problem for linear recurrence sequences, and is a special case of the non-reachability problem for linear time-invariant systems. Our main contribution is a decision procedure for the Positivity Problem for recurrences of order two. The termination proof of our procedure relies on a new transcendence result for infinite series that is of independent interest.
title	Positivity of Nearly Linearly Recurrent Sequences
topic	Dynamical Systems Logic in Computer Science F.4.3
url	https://arxiv.org/abs/2508.00944

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