Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials

Gago-Vargas, Jesús; Hartillo-Hermoso, M. I.; Ucha-Enrı́quez, José Marı́a

doi:10.1016/j.jsc.2005.05.004

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…. • f m Using the results from [13], which we confirmed through intensive testing (cf. [20]), it follows, that the method by Briançon-Maisonobe is the most effective one for the computation of s-parametric annihilators where f = f 1 .…”

Section: The Annihilator Up To Degree Ksupporting

confidence: 57%

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: The Annihilator Up To Degree Ksupporting

confidence: 57%

Constructive D-Module Theory with Singular

et al. 2010

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“…Step 1 of this algorithm was given by Oaku and Takayama (1999); one can also use an alternative method introduced by Briançon and Maisonnobe (2002). See also Ucha-Enríquez and Castro-Jiménez (2004), Gago-Vargas et al (2005), and Levandovskyy and Martin Morales (2008). Steps 2 and 3 were introduced by Oaku (1997b, pp.…”

Section: Note Thatmentioning

confidence: 99%

“…Step 1 should be improved by adopting the alternative method of Briançon and Maisonnobe (2002), as was suggested by Ucha-Enríquez and Castro-Jiménez 2004and Gago-Vargas et al (2005). This would require computations in a ring of differential difference operators, which are not yet available with Risa/Asir.…”

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confidence: 99%

Local Bernstein–Sato ideals: Algorithm and examples

Bahloul

Oaku

2010

Journal of Symbolic Computation

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Let k be a field of characteristic 0. Given a polynomial mapping f = (f 1 ,. .. , f p) from k n to k p , the local Bernstein-Sato ideal of f at a point a ∈ k n is defined as an ideal of the ring of polynomials in s = (s 1 ,. .. , s p). We propose an algorithm for computing local Bernstein-Sato ideals by combining Gröbner bases in rings of differential operators with primary decomposition in a polynomial ring. It also enables us to compute a constructible stratification of k n such that the local Bernstein-Sato ideal is constant along each stratum. We also present examples, some of which have nonprincipal Bernstein-Sato ideals, computed with our algorithm by using the computer algebra system Risa/Asir.

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“…Comparing the effectiveness of the algorithms, Gago-Vargas et al (2005) concluded that the method of Briançon-Maisonobe is the best for the computation of s-parametric annihilators. In Levandovskyy and Martín-Morales (2008) we gave experimental results for the case f = f 1 and showed, that the algorithm of Briançon-Maisonobe is faster than the LOT method, which in turn is faster than the algorithm of Oaku and Takayama.…”

Section: Bernstein-sato Ideals Formentioning

confidence: 99%

Effective Methods for the Computation of Bernstein-Sato polynomials for Hypersurfaces and Affine Varieties

Andres,

Levandovskyy,

Martín-Morales

2010

Preprint

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This paper is the widely extended version of the publication, appeared in Proceedings of IS-SAC'2009 conference (Andres, Levandovskyy, andMartín-Morales, 2009). We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in Singular. Among other, we use this algorithm in computational D-module theory for computing e. g. the Bernstein-Sato polynomial of a single polynomial with several approaches. We also present a new method, having no analogues yet, for the computation of the Bernstein-Sato polynomial of an affine variety. Also, we provide a new proof of the algorithm by Briançon-Maisonobe for the computation of the s-parametric annihilator of a polynomial. Moreover, we present new methods for the latter computation as well as optimized algorithms for the computation of Bernstein-Sato polynomial in various settings.

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Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials

Cited by 5 publications

References 11 publications

Constructive D-Module Theory with Singular

Constructive D-Module Theory with Singular

Local Bernstein–Sato ideals: Algorithm and examples

Effective Methods for the Computation of Bernstein-Sato polynomials for Hypersurfaces and Affine Varieties

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