Un th\'eor\`eme \`a la "Thom-Sebastiani" pour les int\'egrales-fibres

TL;DR

This paper proves a Thom-Sebastiani type theorem for the asymptotics of fiber-integrals of sum functions, relating them to the asymptotics of individual functions and explicitly computing convolution constants.

Contribution

It establishes a precise asymptotic relation for fiber-integrals of sum functions and computes the convolution constants as rational fractions of Gamma functions.

Findings

Asymptotics of fiber-integrals of f⊕g are expressed in terms of individual asymptotics.

Constants in the asymptotic expansion are given by rational fractions of Gamma functions.

Non-vanishing of expected singular terms is demonstrated.

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 $f \oplus g : (x,y) \to f(x) + g(y)$ \ on $(\mathbb{C}^p\times \mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (\mathbb{C}^p,0) \to (\mathbb{C},0)$ and $g : (\mathbb{C}^q,0) \to (\mathbb{C},0)$. 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 theorem, giving the non vanishing of expected singular terms in the asymptotic expansion of $f\oplus g$, 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.