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.
Existence of oblique polar lines for the meromorphic extension of the current valued function $int |f|^{2lambda}|g|^{2mu}square$ is given under the following hypotheses: $f$ and $g$ are holomorphic function germs in $CC^{n+1}$ such that $g$ is non-si
ngular, the germ $S:=ens{d fwedge d g =0}$ is one dimensional, and $g|_S$ is proper and finite. The main tools we use are interaction of strata for $f$ (see cite{B:91}), monodromy of the local system $H^{n-1}(u)$ on $S$ for a given eigenvalue $exp(-2ipi u)$ of the monodromy of $f$, and the monodromy of the cover $g|_S$. Two non-trivial examples are completely worked out.
We prove the following two results 1. For a proper holomorphic function $ f : X to D$ of a complex manifold $X$ on a disc such that ${df = 0 } subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric (a,b)-module $E
^p$ associated to the (filtered) Gauss-Manin connexion of $f$. This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module $E$ we give an integer $N(E)$, explicitely given from simple invariants of $E$, such that the isomorphism class of $Ebig/b^{N(E)}.E$ determines the isomorphism class of $E$. This second result allows to cut asymptotic expansions (in powers of $b$) of elements of $E$ without loosing any information.
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.
In this article we show that all results proved for a large class of holomorphic germs $f : (mathbb{C}^{n+1}, 0) to (mathbb{C}, 0)$ with a 1-dimension singularity in [B.II] are valid for an arbitrary such germ.
