Démonstration constructive de l'existence de polynômes de Bernstein–Sato pour plusieurs fonctions analytiques

Bahloul, Rouchdi

doi:10.1112/s0010437x0400096x

Cited by 20 publications

(26 citation statements)

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“…The existence of non-zero Bernstein-Sato ideals B F has been proved by Sabbah [39], see also Bahloul [3] and Gyoja [21]. In the one-variable case r = 1, the monic generator of the ideal B F is the classical Bernstein-Sato polynomial.…”

Section: Introductionmentioning

confidence: 99%

Bernstein-Sato ideals and local systems

Budur¹

2015

Ann. inst. Fourier

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The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of Malgrange and Kashiwara which states that the Bernstein-Sato polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor fibers of the hypersurface. We also address a multi-variable version of the Monodromy Conjecture, prove that it follows from the usual single-variable Monodromy Conjecture, and prove it in the case of hyperplane arrangements.

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Section: Introductionmentioning

confidence: 99%

Bernstein-Sato ideals and local systems

Budur¹

2015

Ann. inst. Fourier

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“…Moreover, our algorithmic results in the differential case enable the computation of the local Bernstein-Sato polynomial following the results in [4]. Let us notice that the local Bernstein-Sato polynomial constructed in [4] is a product of local b-functions associated with the 1-dimensional cones of the local Gröbner fan of some ideal restricted to a subspace, say V , that lies in the border of W loc .…”

Section: Interest and Applicationsmentioning

confidence: 99%

“…Let us notice that the local Bernstein-Sato polynomial constructed in [4] is a product of local b-functions associated with the 1-dimensional cones of the local Gröbner fan of some ideal restricted to a subspace, say V , that lies in the border of W loc . In [5], we computed an example where the global Gröbner fan restricted to V is trivial, but the local Gröbner fan restricted to V is not.…”

Section: Interest and Applicationsmentioning

confidence: 99%

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Local Gröbner fans

Bahloul

Takayama

2007

Comptes Rendus. Mathématique

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“…In Bahloul's paper [4], this step is made by constructive methods which do not use the flatness property. In this section we will summarize the main results of [3].…”

Section: C{x W }[∆] Consisting Of All Linear Combinations With Polymentioning

confidence: 99%

A Flatness Property for Filtered $\mathcal D$-modules

Castro-Jiménez

Granger

2007

Publ. Res. Inst. Math. Sci.

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Let M be a coherent module over the ring D X of linear differential operators on an analytic manifold X and let Z 1 , . . . , Z k be k germs of transverse hypersurfaces at a point x ∈ X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at x, give rise to a multifiltration U • (M) of M x as in Sabbah's paper [9] and to an analytic standard fan in a way similar to [3]. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9], for which the use of

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Démonstration constructive de l'existence de polynômes de Bernstein–Sato pour plusieurs fonctions analytiques

Abstract: In 1987, Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this paper is to give a more elementary and constructive proof of Sabbah's result based on the notion of the analytic Gröbner fan of a D-module.

Cited by 20 publications

References 3 publications

Bernstein-Sato ideals and local systems

Bernstein-Sato ideals and local systems

Local Gröbner fans

A Flatness Property for Filtered $\mathcal D$-modules

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