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 $(mathbb{C}^ptimes mat
hbb{C}^q, (0,0))$ 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 $foplus g$, 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.
The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple abstract algebraic structure, but very rich, whose prototype is the formal completion of th
e Brieskorn-module of an isolated singularity. The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some finite order jet of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply $b^{-1}.a simeq partial_z.z$ in order to reach a simple pole (a,b)-module. The second invariant is the width which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of $b^{-1}.a$ (the logarithm of the monodromy). In the computation of examples this theorem is quite helpfull because it tells you at which power of $b$ in the expansions you may stop without loosing any information.
