Apparent singularities of D-finite systems

Shaoshi Chen; Manuel Kauers; Ziming Li; Yi Zhang

doi:10.1016/j.jsc.2019.02.009

Apparent singularities of D-finite systems

Shaoshi Chen
, Manuel Kauers
, Ziming Li
, Yi Zhang^*

^*Corresponding author for this work

CAS - Academy of Mathematics and System Sciences
University of Chinese Academy of Sciences
Johannes Kepler University Linz
Austrian Academy of Sciences
University of Texas at Dallas

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.

Original language	English
Pages (from-to)	217-237
Number of pages	21
Journal	Journal of Symbolic Computation
Volume	95
DOIs	https://doi.org/10.1016/j.jsc.2019.02.009
Publication status	Published - 1 Nov 2019
Externally published	Yes

Keywords

Apparent singularity
D-finite system
Desingularization
Formal power series
Gröbner basis
Ordinary point

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10.1016/j.jsc.2019.02.009

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title = "Apparent singularities of D-finite systems",

abstract = "We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.",

keywords = "Apparent singularity, D-finite system, Desingularization, Formal power series, Gr{\"o}bner basis, Ordinary point",

author = "Shaoshi Chen and Manuel Kauers and Ziming Li and Yi Zhang",

note = "Funding Information: In this work, S. Chen was supported in part by the NSFC grants (No. 11501552, 11688101, 11871067) and by the Frontier Key Project (QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS. M. Kauers was supported in part by the Austrian Science Fund (FWF): F50-04 and Y464-N18. Z. Li was supported in part by the NSFC grants: 11771433, 11871067. Y. Zhang was supported in part by the Austrian Science Fund (FWF): Y464-N18 and P29467-N32. Funding Information: In this work, S. Chen was supported in part by the NSFC grants (No. 11501552 , 11688101 , 11871067 ) and by the Frontier Key Project ( QYZDJ-SSW-SYS022 ) and the Fund of the Youth Innovation Promotion Association, CAS . M. Kauers was supported in part by the Austrian Science Fund (FWF): F50-04 and Y464-N18 . Z. Li was supported in part by the NSFC grants: 11771433 , 11871067 . Y. Zhang was supported in part by the Austrian Science Fund (FWF): Y464-N18 and P29467-N32 . Publisher Copyright: {\textcopyright} 2019 Elsevier Ltd",

year = "2019",

month = nov,

day = "1",

doi = "10.1016/j.jsc.2019.02.009",

language = "English",

volume = "95",

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journal = "Journal of Symbolic Computation",

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T1 - Apparent singularities of D-finite systems

AU - Chen, Shaoshi

AU - Kauers, Manuel

AU - Li, Ziming

AU - Zhang, Yi

N1 - Funding Information: In this work, S. Chen was supported in part by the NSFC grants (No. 11501552, 11688101, 11871067) and by the Frontier Key Project (QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS. M. Kauers was supported in part by the Austrian Science Fund (FWF): F50-04 and Y464-N18. Z. Li was supported in part by the NSFC grants: 11771433, 11871067. Y. Zhang was supported in part by the Austrian Science Fund (FWF): Y464-N18 and P29467-N32. Funding Information: In this work, S. Chen was supported in part by the NSFC grants (No. 11501552 , 11688101 , 11871067 ) and by the Frontier Key Project ( QYZDJ-SSW-SYS022 ) and the Fund of the Youth Innovation Promotion Association, CAS . M. Kauers was supported in part by the Austrian Science Fund (FWF): F50-04 and Y464-N18 . Z. Li was supported in part by the NSFC grants: 11771433 , 11871067 . Y. Zhang was supported in part by the Austrian Science Fund (FWF): Y464-N18 and P29467-N32 . Publisher Copyright: © 2019 Elsevier Ltd

PY - 2019/11/1

Y1 - 2019/11/1

N2 - We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.

AB - We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.

KW - Apparent singularity

KW - D-finite system

KW - Desingularization

KW - Formal power series

KW - Gröbner basis

KW - Ordinary point

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DO - 10.1016/j.jsc.2019.02.009

M3 - Article

AN - SCOPUS:85061795884

SN - 0747-7171

VL - 95

SP - 217

EP - 237

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

ER -

Apparent singularities of D-finite systems

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Keywords

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