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Constructive D-Module Theory with Singular

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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.

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Author notes

  1. Jorge Martín-Morales

    Present address: Centro Universitario de la Defensa de Zaragoza, Academia General Militar. Ctra. de Huesca s/n., 50090, Zaragoza, Spain

Authors and Affiliations

  1. Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062, Aachen, Germany

    Daniel Andres & Viktor Levandovskyy

  2. Mathematisches Forschungsinstitut Oberwolfach, Schwarzwaldstr. 9-11, 77709, Oberwolfach-Walke, Germany

    Michael Brickenstein

  3. Department of Mathematics, I.U.M.A., University of Zaragoza, C/Pedro Cerbuna, 12, 50009, Saragossa, Spain

    Jorge Martín-Morales

  4. Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str. 48, 67632, Kaiserslautern, Germany

    Hans Schönemann

Authors
  1. Daniel Andres
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  2. Michael Brickenstein
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  3. Viktor Levandovskyy
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  4. Jorge Martín-Morales
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  5. Hans Schönemann
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Correspondence to Viktor Levandovskyy.

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Andres, D., Brickenstein, M., Levandovskyy, V. et al. Constructive D-Module Theory with Singular . Math.Comput.Sci. 4, 359–383 (2010). https://doi.org/10.1007/s11786-010-0058-x

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  • DOI: https://doi.org/10.1007/s11786-010-0058-x

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