An Efficient Modular Algorithm for Computing the Global B-Function

Noro, Masayuki

doi:10.1142/9789812777171_0015

Cited by 19 publications

(27 citation statements)

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“…For an efficient computation of in (−w,w) (I f ) using the method of weighted homogenization as described in Section 5.1, Noro proposes [30] to choose the weightsû = (deg u (f ), u 1 , . .…”

Section: Global Bernstein-sato Polynomialmentioning

confidence: 99%

“…There are several implementations of algorithms for D-modules, namely the experimental program kan/sm1 by N. Takayama [38], the bfct package in Risa/Asir [31] by M. Noro [30] and the package Dmodules.m2 in Macaulay2 by A. Leykin and H. Tsai [40]. We aim at creating a D-module suite, which will combine flexibility and rich functionality with high performance, being able to treat more complicated examples.…”

Section: Introductionmentioning

confidence: 99%

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Constructive D-Module Theory with Singular

et al. 2010

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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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Section: Global Bernstein-sato Polynomialmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Constructive D-Module Theory with Singular

et al. 2010

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“…, ∂ n . This trick was used for the first time by Noro in [22], where a modular method to speed up b-function computations is provided as well. We include the following algorithm for the convenience of the reader as a similar syzygetic approach will be used in Algorithms 3.2, 4.5, and 5.12.…”

Section: Global Bernstein-sato Polynomialsmentioning

confidence: 99%

Algorithms for Bernstein--Sato polynomials and multiplier ideals

Berkesch

Leykin

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After providing a survey of known algorithms for computing the global b-function, we develop a new method to compute the local b-function for a single polynomial. We then develop algorithms that compute generalized Bernstein-Sato polynomials of Budur-Mustaţǎ-Saito and Shibuta for an arbitrary polynomial ideal. These lead to computations of log canonical thresholds, jumping coefficients, and multiplier ideals. Our algorithm for multiplier ideals simplifies that of Shibuta and shares a common subroutine with our local b-function algorithm. The algorithms we present have been implemented in the D-modules package of the computer algebra system Macaulay2.

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“…Our code is merged to the D-module package [8] with the command name logCohomology. Unfortunately, this implementation has not installed an efficient algorithm of computing b-function by Noro [12] to get the truncated complex in [15,17]. Then, only relatively small examples are feasible.…”

Section: Implementation and Examplesmentioning

confidence: 99%

“…Example 4.1 is computed by our Macaulay2 program. Example 4.2 is computed by our implementation on kan/k0 (logc2.k, http://www.openxm.org) and Risa/Asir with an implementation of [12] (the transfer algorithm has not been implemented yet for kan/k0). Our implementation does not contain that for the Quillen-Suslin theorem.…”

Section: Implementation and Examplesmentioning

confidence: 99%