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Daniel Barlet
Un théorème à la « Thom-Sebastiani » pour les intégrales-fibres
(A Thom-Sebastiani theorem for fiber-integrals)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 319-353, doi: 10.5802/aif.2524
Article PDF | Reviews MR 2664317 | Zbl 1194.32015
Class. Math.: 32S25, 32S40, 32S50
Keywords: Asymptotic expansions, fiber-integrals, Thom-Sebastiani theorem

Résumé - Abstract

The aim of this article is to prove a Thom-Sebastiani theorem for the asymptotics of the fiber-integrals. This means that we describe the asymptotics of the fiber-integrals of the function $f \oplus g : (x,y) \rightarrow f(x) + g(y)$ on $(\mathbb{C}^p\times \mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (\mathbb{C}^p,0) \rightarrow (\mathbb{C},0)$ and $g : (\mathbb{C}^q,0) \rightarrow (\mathbb{C},0)$. This reduces to compute the asymptotics of a convolution $\Phi *\Psi $ from the asymptotics of $\Phi $ and $\Psi $ modulo smooth terms.

To obtain a precise result, giving the non vanishing of expected singular terms in the asymptotic expansions of the fiber-integrals associated to $f\oplus g$, we have to compute the constants coming from the convolution process. We show that they are given by rational fractions of Gamma factors. This enable us to show that these constants do not vanish.

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