LOGARITHMIC COMPARISON THEOREM AND <i>D</i>-MODULES: AN OVERVIEW

Torrelli, Tristan

doi:10.1142/9789812707499_0040

Cited by 11 publications

(14 citation statements)

References 27 publications

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“…Let us see an example that could not be treated before with the classical methods. This example was suggested by F. Castro-Jiménez and J.-M. Ucha for testing the Logarithmic Comparison Theorem, see e. g. [34].…”

Section: Minimal Integral Root Of B F (S) and The Logarithmic Compari...mentioning

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: Minimal Integral Root Of B F (S) and The Logarithmic Compari...mentioning

confidence: 99%

“…suggested by F. Castro-Jiménez and J.-M. Ucha for testing the Logarithmic Comparison Theorem, see e. g [34]…”

mentioning

confidence: 99%

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

Levandovskyy

Martín-Morales

2012

Journal of Algebra

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“…This example was suggested by F. Castro-Jiménez and J. M. Ucha for testing the Logarithmic Comparison Theorem. A nice introduction to this topic can be found, for instance, in [39].…”

Section: Partial Knowledge Of Bernstein-sato Polynomialmentioning

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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“…One expects the analytic Logarithmic Comparison Theorem to inform certain D X -constructions, especially Bernstein-Sato polynomials. See the surveys [34], [22]. For some free divisors there is an intrinsic D X -theoretic formulation of the Logarithmic Comparison Theorem as developed in [8]; see also [23] for current developments.…”

Section: Introductionmentioning

confidence: 99%

Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr{D}_{X}$-modules

Bath¹

2022

Preprint

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For a reduced hyperplane arrangement we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasiisomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement's complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao and Yuzvinsky. We also prove that: every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite dimensional linear algebra. Finally, we give some D X -module applications: for example, we give a sharp restriction on the codimension one components of the multivariate Bernstein-Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito's result that the roots of the Bernstein-Sato polynomial of a non-smooth, central, reduced arrangement live in (−2 + 1/d, 0).

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LOGARITHMIC COMPARISON THEOREM AND D-MODULES: AN OVERVIEW

Cited by 11 publications

References 27 publications

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

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

Constructive D-Module Theory with Singular

Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr{D}_{X}$-modules

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