Article

Differential equations for algebraic functions

Authors:

Frédéric Chyzak,

Grégoire Lecerf,

Éric SchostAuthors Info & Claims

ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation

Pages 25 - 32

https://doi.org/10.1145/1277548.1277553

Published: 29 July 2007 Publication History

Abstract

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series.

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Cited By

van der Hoeven JLecerf G(2024)Plane curve germs and contact factorizationApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00669-zOnline publication date: 30-Nov-2024
https://doi.org/10.1007/s00200-024-00669-z
Bostan ANeiger VYurkevich S(2023)Beating binary powering for polynomial matricesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597118(70-79)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597118
Bell JLiu HMishna M(2023)Cogrowth series for free products of finite groupsInternational Journal of Algebra and Computation10.1142/S021819672350013333:02(237-260)Online publication date: 14-Jan-2023
https://doi.org/10.1142/S0218196723500133
Show More Cited By

Index Terms

Differential equations for algebraic functions
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

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cover image ACM Conferences

ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation

July 2007

406 pages

ISBN:9781595937438

DOI:10.1145/1277548

General Chair:
Dongming Wang
Beihang University and UPMC-CNRS (China/France)

Copyright © 2007 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 29 July 2007

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July 29 - August 1, 2007

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

van der Hoeven JLecerf G(2024)Plane curve germs and contact factorizationApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00669-zOnline publication date: 30-Nov-2024
https://doi.org/10.1007/s00200-024-00669-z
Bostan ANeiger VYurkevich S(2023)Beating binary powering for polynomial matricesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597118(70-79)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597118
Bell JLiu HMishna M(2023)Cogrowth series for free products of finite groupsInternational Journal of Algebra and Computation10.1142/S021819672350013333:02(237-260)Online publication date: 14-Jan-2023
https://doi.org/10.1142/S0218196723500133
Bostan ARivoal TSalvy B(2023)Minimization of differential equations and algebraic values of 𝐸-functionsMathematics of Computation10.1090/mcom/3912Online publication date: 23-Oct-2023
https://doi.org/10.1090/mcom/3912
Kauers MKauers M(2023)Summation and IntegrationD-Finite Functions10.1007/978-3-031-34652-1_5(393-509)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_5
Kauers MKauers M(2023)OperatorsD-Finite Functions10.1007/978-3-031-34652-1_4(287-391)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_4
Kauers MKauers M(2023)The Differential Case in One VariableD-Finite Functions10.1007/978-3-031-34652-1_3(185-286)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_3
Kauers MKauers M(2023)The Recurrence Case in One VariableD-Finite Functions10.1007/978-3-031-34652-1_2(81-183)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_2
Kauers MKauers M(2023)Background and Fundamental ConceptsD-Finite Functions10.1007/978-3-031-34652-1_1(1-80)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_1
Huang HKauers MMukherjee GMaza MZhi L(2022)Order-Degree-Height Surfaces for Linear OperatorsProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3536187(91-99)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3536187
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