skip to main content
10.1145/860854.860887acmconferencesArticle/Chapter ViewAbstractPublication PagesissacConference Proceedingsconference-collections
Article

Shiftless decomposition and polynomial-time rational summation

Published:03 August 2003Publication History

ABSTRACT

New algorithms are presented for computing the dispersion set of two polynomials over Q and for shiftless factorization. Together with a summability criterion by Abramov, these are applied to get a polynomial-time algorithm for indefinite rational summation, using a sparse representation of the output.

References

  1. S.A. Abramov. On the summation of rational functions. U.S.S.R. Comput. Maths. Math. Phys. 11, pp. 324--330, 1971. Transl. from Zh. vychisl. mat. mat. fiz. 11, pp. 1071--1075, 1971.Google ScholarGoogle ScholarCross RefCross Ref
  2. S.A. Abramov. The rational component of the solution of a first-order linear recurrence relation with a rational right-hand side. U.S.S.R. Comput. Maths. Math. Phys. 15, pp. 216--221, 1975. Transl. from Zh. vychisl. mat. mat. fiz. 15, pp. 1035--1039, 1975.Google ScholarGoogle ScholarCross RefCross Ref
  3. S.A. Abramov. Indefinite sums of rational functions. Proceedings ISSAC'95, pp. 303--308. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. E. Bach, J. Driscoll and J. O. Shallit. Factor Refinement. Journal of Algorithms. 15, pp. 199--222, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. E. Bach and J. Shallit. Algorithmic Number Theory. Volume 1: Efficient Algorithms. MIT Press, Boston MA, 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, Cambridge, U.K., 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. J. Gerhard. Modular algorithms in symbolic summation and symbolic integration. PhD thesis, Universität Paderborn, Germany, 2001.Google ScholarGoogle Scholar
  8. R.W. Gosper. Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75(1), pp. 40--42, 1978.Google ScholarGoogle ScholarCross RefCross Ref
  9. M. van Hoeij. Rational solutions of linear difference equations. Proceedings ISSAC'98, pp. 120--123, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. M. van Hoeij. Factoring polynomials and the knapsack problem. J. Number Theory. 95, pp. 167-189, 2002.Google ScholarGoogle ScholarCross RefCross Ref
  11. P. Lisonek, P. Paule and V. Strehl. Improvement of the Degree Setting in Gosper's Algorithm. J. Symbolic Comput. 16, pp. 243--258, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Y.K Man. On computing closed forms for indefinite summation. J. Symbolic Comput. 16, pp. 355--376, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Y.K. Man and F.J. Wright. Fast polynomial dispersion computation and its application to indefinite summation. Proceedings ISSAC'94, pp. 175--180, 1994. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. J.B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, pp. 64--94, 1962.Google ScholarGoogle ScholarCross RefCross Ref
  15. P. Paule. Greatest factorial factorization and symbolic summation. J. Symbolic Comput. 20(3), pp. 235--268, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. R. Pirastu. On combinatorial identities: symbolic summation and umbral calculus. PhD thesis, Johannes Kepler Universität Linz, Austria, July 1996.Google ScholarGoogle Scholar

Index Terms

  1. Shiftless decomposition and polynomial-time rational summation

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in
    • Published in

      cover image ACM Conferences
      ISSAC '03: Proceedings of the 2003 international symposium on Symbolic and algebraic computation
      August 2003
      284 pages
      ISBN:1581136412
      DOI:10.1145/860854
      • General Chair:
      • Hoon Hong

      Copyright © 2003 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]

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 3 August 2003

      Permissions

      Request permissions about this article.

      Request Permissions

      Check for updates

      Qualifiers

      • Article

      Acceptance Rates

      ISSAC '03 Paper Acceptance Rate36of68submissions,53%Overall Acceptance Rate395of838submissions,47%

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader