Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors

Macarro, L. Narváez

doi:10.1090/conm/474/09259

Cited by 8 publications

(3 citation statements)

References 27 publications

(73 reference statements)

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“…and thus after computing b 2 ) be the last example from [22]. The Bernstein-Sato polynomial and the polynomial β(s) are respectively b f (s) = (s + 1) 3 (s + 3/4)(s + 3/8)(s + 9/8)(s + 1/4)(s + 7/8)(s + 1/2)(s + 5/8), β(s) = (s + 3/4)(s + 5/8)(s + 1/2)(s + 3/8)(s + 1/4).…”

Section: Stratification Associated With Local B-functionsmentioning

confidence: 99%

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Algorithms for checking rational roots of b-functions and their applications

Levandovskyy

Martín-Morales

2012

Journal of Algebra

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Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called checkRoot for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and b-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A'Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications consists of computing of a stratification of affine space with the local b-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Also we apply our results for the computation of Bernstein-Sato polynomials for varieties. The methods from this paper have been implemented in Singular:Plural as libraries dmod.lib and bfun.lib. All the examples from the paper have been computed with this implementation. (−α), . . . P k (−α), f . Since J = D[s] ⇐⇒ J ∩ K[s] = K[s] ⇐⇒ K = D n , and gcd(b f (s), s + α) = 1 if and only if b f (−α) = 0, the result follows from applying Theorem 2.1 using q(s) = s + α.Once we know a system of generators of the annihilator of f s in D n [s], the last corollary provides an algorithm for checking whether a given rational number is a root of the b-function of f , using Gröbner bases in the Weyl algebra.a system of generators of Ann Dn[s] (f s ); Input 2: f , a polynomial in R n ; α, a number in Q >0 ; Output: true, if α is a root of b f (−s); false, otherwise; K := P 1 (−α), . . . , P k (−α), f ; K = J ∩ D n ⊆ D n G := reduced Gröbner basis of K w.r.t. ANY term ordering; return (G = {1}); MultiplicitiesTwo approaches to deal with multiplicities are presented. We start with a natural generalization of Corollary 2.3. Corollary 2.4. Let m α be the multiplicity of α as a root of b f (−s) and let us consider the ideals J i = Ann Dn[s] (f s ) + f, (s + α) i+1 ⊆ D n [s], i = 0, . . . , n. The following conditions are equivalent:Moreover if D n [s] J 0 J 1 · · · J m−1 = J m , then m α = m. In particular, m ≤ n and J m−1 = J m = · · · = J n .Proof. 1 ⇐⇒ 2. Since m α > i if and only if gcd(b f (s), (s+α) i+1 ) = (s+α) i+1 , the equivalence follows by applying Theorem 2.1 (1) using q(s) = (s + α) i+1 . 2 =⇒ 3. If (s + α) i ∈ J i ∩ K[s], then clearly J i ∩ K[s] (s + α) i+1 .

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Section: Stratification Associated With Local B-functionsmentioning

confidence: 99%

“…(1) D[s] (f s ). For all the examples treated in [22], he was able to compute an operator P (s) ∈ D[s] such that b f (s) − P (s)f ∈ Ann (1) D[s] (f s ). The last example in the paper is quite involved and could not be computed by using any computer algebra system directly.…”

Section: Other Applicationsmentioning

confidence: 99%

Algorithms for checking rational roots of b-functions and their applications

Levandovskyy

Martín-Morales

2012

Journal of Algebra

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“…More generally, for a given k ≥ 1 one can consider the left ideal Ann has been recently studied by Narváez in [28]. It is an open problem to find the minimal integer k 0 satisfying the above condition without computing the whole annihilator.…”

Section: The Annihilator Up To Degree Kmentioning

confidence: 99%

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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