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Order bounds for C2-finite sequences

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Veronika PillweinAuthors Info & Claims

ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation

Pages 389 - 397

https://doi.org/10.1145/3597066.3597070

Published: 24 July 2023 Publication History

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Abstract

A sequence is called C-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with C-finite coefficients. Recently, it was shown that such C2-finite sequences satisfy similar closure properties as C-finite sequences. In particular, they form a difference ring.

In this paper we present new techniques for performing these closure properties of C2-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations.

The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.

References

[1]

Jean Berstel and Maurice Mignotte. 1976. Deux propriétés décidables des suites récurrentes linéaires. Bulletin de la Société Mathématique de France 104 (1976), 175–184.

[2]

P. B. Bhattacharya, S. K. Jain, and S. R. Nagpaul. 1994. Basic Abstract Algebra (2 ed.). Cambridge University Press.

[3]

Richard P. Brent. 1976. Fast Multiple-Precision Evaluation of Elementary Functions. J. ACM 23, 2 (1976), 242–251.

Digital Library

[4]

Cristina Caldeira and João Filipe Queiró. 2015. Invariant factors of products over elementary divisor domains. Linear Algebra Appl. 485 (2015), 345–358.

[5]

David Carlson and Eduardo Marques De Sá. 1984. Generalized minimax and interlacing theorems. Linear and Multilinear Algebra 15, 1 (1984), 77–103.

[6]

Henri Cohen. 2013. A Course in Computational Algebraic Number Theory. Springer Berlin Heidelberg.

Digital Library

[7]

Charles W. Curtis and Irving Reiner. 1966. Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers.

[8]

Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward. 2015. Recurrence Sequences. American Mathematical Society.

[9]

Paolo Faccin. 2014. Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras. Ph.D. Dissertation. University of Trento.

[10]

Guoqiang Ge. 1993. Algorithms Related to Multiplicative Representations of Algebraic Numbers. Ph.D. Dissertation. U.C. Berkeley.

[11]

Antonio Jiménez-Pastor, Philipp Nuspl, and Veronika Pillwein. 2021. On C2-finite sequences. In Proceedings of ISSAC 2021, Virtual Event Russian Federation, July 18–23, 2021. 217–224.

[12]

Antonio Jiménez-Pastor, Philipp Nuspl, and Veronika Pillwein. 2023. An extension of holonomic sequences: C2-finite sequences. Journal of Symbolic Computation 116 (2023), 400–424.

Digital Library

[13]

Manuel Kauers. 2005. Algorithms for Nonlinear Higher Order Difference Equations. Ph.D. Dissertation. Johannes Kepler University Linz.

[14]

Manuel Kauers. 2013. The Holonomic Toolkit. In Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Springer, 119–144.

[15]

Manuel Kauers. 2014. Bounds for D-Finite Closure Properties. In Proceedings of ISSAC 2014, Kobe, Japan. Association for Computing Machinery, New York, NY, USA, 288–295.

Digital Library

[16]

Manuel Kauers and Peter Paule. 2011. The Concrete Tetrahedron. Springer.

[17]

Tomer Kotek and Johann A. Makowsky. 2014. Recurrence relations for graph polynomials on bi-iterative families of graphs. Eur. J. Comb. 41 (2014), 47–67.

Digital Library

[18]

Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda. 2022. Ansatz in a Nutshell: A comprehensive step-by-step guide to polynomial, C-finite, holonomic, and C2-finite sequences. arxiv:math/2201.08035https://arxiv.org/abs/2201.08035.

[19]

Arjen K. Lenstra, Hendrik W. jun. Lenstra, and László Lovász. 1982. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.

[20]

Christian Mallinger. 1996. Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences. Diplomarbeit. Johannes Kepler University Linz.

[21]

David W. Masser. 1988. Linear relations on algebraic groups. New Advances in Transcendence Theory (1988).

[22]

Stephen Melczer. 2021. An Invitation to Analytic Combinatorics. Springer.

[23]

Johannes Middeke. 2019. Symbolic Linear Algebra. https://www3.risc.jku.at/education/courses/ss2019/sla/script_sla2019.pdf.

[24]

Morris Newman. 1972. Integral Matrices. Elsevier Science.

[25]

Philipp Nuspl. 2022. C-finite and C2-finite Sequences in SageMath. RISC Report Series 22-06. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz.

[26]

Philipp Nuspl and Veronika Pillwein. 2022. A Comparison of Algorithms for Proving Positivity of Linearly Recurrent Sequences. In Computer Algebra in Scientific Computing(LNCS, Vol. 13366). Springer International Publishing, 268–287.

[27]

Philipp Nuspl and Veronika Pillwein. 2022. Simple C2-finite Sequences: a Computable Generalization of C-finite Sequences. In ISSAC ’22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, Marc Moreno Maza and Lihong Zhi (Eds.). Association for Computing Machinery, 45–53.

Digital Library

[28]

OEIS Foundation Inc.2023. The On-Line Encyclopedia of Integer Sequences. http://www.oeis.org.

[29]

Joël Ouaknine and James Worrell. 2012. Decision Problems for Linear Recurrence Sequences. In Lecture Notes in Computer Science. Springer, 21–28.

[30]

Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. 1996. A = B. CRC Press.

[31]

Carsten Schneider. 2020. Minimal Representations and Algebraic Relations for Single Nested Products. Programming and Computer Software 46 (2020), 133–161. Issue 2.

Digital Library

[32]

Richard P. Stanley. 1980. Differentiably Finite Power Series. European Journal of Combinatorics 1, 2 (1980), 175–188.

[33]

Terrence Tao. 2022. Using the Smith normal form to manipulate lattice subgroups and closed torus subgroups. https://terrytao.wordpress.com/2022/07/20/using-the-smith-normal-form-to-manipulate-lattice-subgroups-and-closed-torus-subgroups/. Accessed: 24.4.2023.

[34]

Thotsaporn Aek Thanatipanonda and Yi Zhang. 2020. Sequences: Polynomial, C-finite, Holonomic,...arxiv:math/2004.01370https://arxiv.org/pdf/2004.01370.

[35]

Mark van Hoeij. 2002. Factoring polynomials and the Knapsack problem. Journal of Number Theory 95 (2002), 167–189.

[36]

Mark van Hoeij. 2013. The complexity of factoring univariate polynomials over the rationals. In Proc. ISSAC’13. 13–14.

[37]

Mark van Hoeij and Andrew Novocin. 2012. Gradual Sub-lattice Reduction and a New Complexity for Factoring Polynomials. Algorithmica 63, 3 (2012), 616–633.

Digital Library

[38]

Tao Zheng. 2020. Characterizing Triviality of the Exponent Lattice of a Polynomial Through Galois and Galois-Like Groups. In Computer Algebra in Scientific Computing, François Boulier, Matthew England, Timur M. Sadykov, and Evgenii V. Vorozhtsov (Eds.). Springer International Publishing, 621–641.

[39]

Tao Zheng. 2021. A fast algorithm for computing multiplicative relations between the roots of a generic polynomial. Journal of Symbolic Computation 104 (2021), 381–401.

[40]

Tao Zheng and Bican Xia. 2019. An Effective Framework for Constructing Exponent Lattice Basis of Nonzero Algebraic Numbers. In Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (Beijing, China). 371–378.

Digital Library

Cited By

Moosbrugger MMüllner JBartocci EKovács L(2024)Polar: An Algebraic Analyzer for (Probabilistic) LoopsPrinciples of Verification: Cycling the Probabilistic Landscape10.1007/978-3-031-75783-9_8(179-200)Online publication date: 13-Nov-2024
https://doi.org/10.1007/978-3-031-75783-9_8

Index Terms

Order bounds for C2-finite sequences
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms

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Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred ...
An extension of holonomic sequences: C ²-finite sequences
Abstract
Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients ...
Simple C2-finite Sequences: a Computable Generalization of C-finite Sequences
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A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C2-finite sequences is a natural generalization of holonomic ...

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cover image ACM Other conferences

ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation

July 2023

567 pages

ISBN:9798400700392

DOI:10.1145/3597066

Editors:
Alicia Dickenstein
University of Buenos Aires, Argentina
,
Elias Tsigaridas
INRIA Paris and Sorbonne University, France
,
Gabriela Jeronimo
University of Buenos Aires, Argentina

Copyright © 2023 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 24 July 2023

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Austrian Science Fund

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ISSAC 2023

ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023

July 24 - 27, 2023

Tromsø, Norway

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Cited By

Moosbrugger MMüllner JBartocci EKovács L(2024)Polar: An Algebraic Analyzer for (Probabilistic) LoopsPrinciples of Verification: Cycling the Probabilistic Landscape10.1007/978-3-031-75783-9_8(179-200)Online publication date: 13-Nov-2024
https://doi.org/10.1007/978-3-031-75783-9_8

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