eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-18
67:1
67:15
10.4230/LIPIcs.MFCS.2021.67
article
On Positivity and Minimality for Second-Order Holonomic Sequences
Kenison, George
1
Klurman, Oleksiy
2
3
Lefaucheux, Engel
4
Luca, Florian
5
6
7
Moree, Pieter
3
https://orcid.org/0000-0002-5318-2587
Ouaknine, Joël
4
https://orcid.org/0000-0003-0031-9356
Whiteland, Markus A.
4
https://orcid.org/0000-0002-6006-9902
Worrell, James
8
Institute for Logic and Computation, The Technical University of Vienna, Austria
School of Mathematics, University of Bristol, UK
Max Planck Institute for Mathematics, Bonn, Germany
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
School of Mathematics, University of the, Witwatersrand, Johannesburg, South Africa
Research Group in Algebraic Structures & , Applications, King Abdulaziz University, Riyadh, Saudi Arabia
Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
Department of Computer Science, University of Oxford, UK
An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞.
In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol202-mfcs2021/LIPIcs.MFCS.2021.67/LIPIcs.MFCS.2021.67.pdf
Holonomic sequences
Minimal solutions
Positivity Problem