Caractérisation géométrique de l’existence du polynôme de Bernstein relatif

Briançon, Joël; Maisonobe, Ph.

doi:10.1007/978-3-0348-9020-5_12

Cited by 7 publications

(5 citation statements)

References 5 publications

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“…If p is a smooth point of Var(f ) then f can be used as an analytic coordinate near p, hence b f,p an (s) = s + 1, and so b f (s) = s + 1 for all smooth hypersurfaces. By Proposition 2.6 in [35], an extension of [37], the equation b f (s) = s+1 implies smoothness of Var(f ). Explicit formulae for the Bernstein-Sato polynomial are rare; here are some classes of examples.…”

Section: Special Casesmentioning

confidence: 99%

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Walther

2016

Invent. math.

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Abstract. For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by f s to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of f s is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials.In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1/f ; we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

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Section: Special Casesmentioning

confidence: 99%

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Walther

2016

Invent. math.

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“…This is defined by iterating the mapping cone of ψ x i ,1 → ϕ x i ,1 where the x i are local coordinates. Then the Hodge filtration on the Milnor cohomology is given by F )), see also [3]. So the assertion follows.…”

Section: 5mentioning

confidence: 77%

Multiplier ideals, V-filtrations and transversal sections

et al. 2006

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“…It is clear that any "relative Bernstein-Sato polynomial" is comprehensive but the converse is obviously wrong since a relative Bernstein-Sato polynomial does not exist in general, even when p = 1 (see e.g. [8] for the definition of a relative Bernstein-Sato polynomial, see also [12] for general results on the subject in the hypersurface case). Corollary 3.…”

Section: Resultsmentioning

confidence: 99%

Some results on Bernstein-Sato polynomials for parametric analytic functions

Bahloul¹

2006

Proc. Japan Acad. Ser. A Math. Sci.

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This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic (and under some conditions, a relative) Bernstein-Sato polynomial.

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Caractérisation géométrique de l’existence du polynôme de Bernstein relatif

Cited by 7 publications

References 5 publications

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Multiplier ideals, V-filtrations and transversal sections

Some results on Bernstein-Sato polynomials for parametric analytic functions

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