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Mathematics in Computer Science

, Volume 4, Issue 2–3, pp 359–383 | Cite as

Constructive D-Module Theory with Singular

  • Daniel Andres
  • Michael Brickenstein
  • Viktor Levandovskyy
  • Jorge Martín-Morales
  • Hans Schönemann
Article

Abstract

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein–Sato polynomials and also algorithms, recovering any kind of Bernstein–Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein–Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Keywords

D-modules Non-commutative Gröbner basis Annihilator ideal b-Function Bernstein–Sato polynomial Bernstein–Sato ideal 

Mathematics Subject Classification (2010)

13P10 14F10 68W30 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Daniel Andres
    • 1
  • Michael Brickenstein
    • 2
  • Viktor Levandovskyy
    • 1
  • Jorge Martín-Morales
    • 3
    • 4
  • Hans Schönemann
    • 5
  1. 1.Lehrstuhl D für MathematikRWTH AachenAachenGermany
  2. 2.Mathematisches Forschungsinstitut OberwolfachOberwolfach-WalkeGermany
  3. 3.Department of Mathematics, I.U.M.A.University of ZaragozaSaragossaSpain
  4. 4.Centro Universitario de la Defensa de ZaragozaZaragozaSpain
  5. 5.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany

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