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Equations fonctionnelles pour une fonction sur
un espace singulier

Abstract : Consider an analytic complete intersection X in a complex manifold V, defined by a morphism g. In order to extend to the singular case some results of the theory of Bernstein-Sato polynomials, we study Bernstein polynomials of an analytic function f on V associated to sections of the local cohomology module R with support in X. Indeed, it follows from the algebraic construction of vanishing cycles that the roots of these polynomials are intimately connected to eigenvalues of the local monodromy of f on X.

We begin with some results on Bernstein polynomials associated to sections of a holonomic D-Module. Then we study the cases of X smooth, and of f smooth, X a hypersurface. Next, we study the existence of generic and relative Bernstein polynomials
associated to sections of R and attached to an analytic
deformation. We link these problems to the geometry of conormal spaces.

Following ideas of B. Malgrange, we give a construction for studying
Bernstein polynomials associated to sections of R when the morphisms g and (f,g) define isolated complete intersection singularities. Our construction needs g to be the weighted homogeneous, and it requires computations of annihilators. Finally, we describe several computations which use this result. First, we
give an algorithm to compute Bernstein polynomials if in addition to the usual hypotheses, we suppose that the initial form of f defines an isolated singularity on X. When f is in fact quasi-homogeneous, we obtain explicit formulas. We end this work with complete computations when X is a nondegenerate quadratic cone.
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Contributor : Tristan Torrelli <>
Submitted on : Friday, December 23, 2005 - 1:50:34 PM
Last modification on : Monday, October 12, 2020 - 10:27:29 AM
Long-term archiving on: : Saturday, April 3, 2010 - 7:51:33 PM



  • HAL Id : tel-00011262, version 1



Tristan Torrelli. Equations fonctionnelles pour une fonction sur
un espace singulier. Mathématiques [math]. Université Nice Sophia Antipolis, 1998. Français. ⟨tel-00011262⟩



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