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Fast algorithms for differential equations in positive characteristic

Authors:

Éric SchostAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 47 - 54

https://doi.org/10.1145/1576702.1576712

Published: 28 July 2009 Publication History

Abstract

We address complexity issues for linear differential equations in characteristic p >;0: resolution and computation of the p-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to p. We prove bounds linear in p on the degree of polynomial solutions and propose algorithms for testing the existence of polynomial solutions in sublinear time Õ(p^1/2), and for determining a whole basis of the solution space in quasi-linear time Õ(p); the Õ notation indicates that we hide logarithmic factors. We show that for equations of arbitrary order, the p-curvature can be computed in subquadratic time Õ(p^1.79), and that this can be improved to O(log(p)) for first order equations and to Õ(p) for classes of second order equations.

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Cited By

Bostan ACaruso XRoques J(2024)Algebraic solutions of linear differential equations: An arithmetic approachBulletin of the American Mathematical Society10.1090/bull/183561:4(609-658)Online publication date: 15-Aug-2024
https://doi.org/10.1090/bull/1835
Bostan AYurkevich S(2023)Fast computation of the N-th term of a q-holonomic sequence and applicationsJournal of Symbolic Computation10.1016/j.jsc.2022.07.008115:C(96-123)Online publication date: 1-Mar-2023
https://dl.acm.org/doi/10.1016/j.jsc.2022.07.008
Di Vizio LHardouin CGranier A(2022)Intrinsic Approach to Galois Theory of 𝑞-Difference EquationsMemoirs of the American Mathematical Society10.1090/memo/1376279:1376Online publication date: Sep-2022
https://doi.org/10.1090/memo/1376
Show More Cited By

Index Terms

Fast algorithms for differential equations in positive characteristic
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

Copyright © 2009 ACM.

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Published: 28 July 2009

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ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Bostan ACaruso XRoques J(2024)Algebraic solutions of linear differential equations: An arithmetic approachBulletin of the American Mathematical Society10.1090/bull/183561:4(609-658)Online publication date: 15-Aug-2024
https://doi.org/10.1090/bull/1835
Bostan AYurkevich S(2023)Fast computation of the N-th term of a q-holonomic sequence and applicationsJournal of Symbolic Computation10.1016/j.jsc.2022.07.008115:C(96-123)Online publication date: 1-Mar-2023
https://dl.acm.org/doi/10.1016/j.jsc.2022.07.008
Di Vizio LHardouin CGranier A(2022)Intrinsic Approach to Galois Theory of 𝑞-Difference EquationsMemoirs of the American Mathematical Society10.1090/memo/1376279:1376Online publication date: Sep-2022
https://doi.org/10.1090/memo/1376
Bostan AEmiris IZhi LMantzaflaris A(2020)Computing the N-th term of a q-holonomic sequenceProceedings of the 45th International Symposium on Symbolic and Algebraic Computation10.1145/3373207.3404060(46-53)Online publication date: 20-Jul-2020
https://dl.acm.org/doi/10.1145/3373207.3404060
Hyun SLebreton RSchost ÉYap CEl Din M(2017)Algorithms for Structured Linear Systems Solving and Their ImplementationProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087659(205-212)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087659
Bostan ACaruso XSchost ÉAbramov SZima EGao X(2016)Computation of the Similarity Class of the p-CurvatureProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930897(111-118)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930897
Bostan ACaruso XSchost ÉRobertz DLinton SYokoyama K(2015)A Fast Algorithm for Computing the P-curvatureProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756674(69-76)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756674
Bostan ACaruso XSchost ÉNagasaka KWinkler FSzanto A(2014)A fast algorithm for computing the characteristic polynomial of the p-curvatureProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608650(59-66)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608650
Bostan ASalvy BChowdhury MSchost ÉLebreton Rvan der Hoeven Jvan Hoeij M(2012)Power series solutions of singular (q)-differential equationsProceedings of the 37th International Symposium on Symbolic and Algebraic Computation10.1145/2442829.2442848(107-114)Online publication date: 22-Jul-2012
https://dl.acm.org/doi/10.1145/2442829.2442848
Banderier CHitczenko P(2012)Enumeration and asymptotics of restricted compositions having the same number of partsDiscrete Applied Mathematics10.1016/j.dam.2011.12.011160:18(2542-2554)Online publication date: 1-Dec-2012
https://dl.acm.org/doi/10.1016/j.dam.2011.12.011

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