  We remark that the study of a fiber-integral of the type F (s) := f =s
($\omega$/df) $\land$ ($\omega$/df) either in the local case where $\rho$
$\not\equiv$ 1 around 0 is C $\infty$ and compactly supported near the origin
which is a singular point of {f = 0} in C n+1 , or in a global setting where f
: X $\rightarrow$ D is a proper holomorphic function on a complex manifold X,
smooth outside {f = 0} with $\rho$ $\not\equiv$ 1 near {f = 0}, for given
holomorphic (n+1)--forms $\omega$ and $\omega$' , that a better control on the
asymptotic expansion of F when s $\rightarrow$ 0, is obtained by using the
Bernstein polynomial of the "frescos" associated to f and $\omega$ and to f and
$\omega$' (a fresco is a "small" Brieskorn module corresponding to the
differential equation deduced from the Gauss-Manin system of f at 0) than to
use the Bernstein polynomial of the full Gauss-Manin system of f at the origin.
We illustrate this in the local case in some rather simple (non
quasi-homogeneous) polynomials, where the Bernstein polynomial of such a fresco
is explicitly evaluate. AMS Classification. 32 S 25, 32 S 40. Key words.
Fiber-integrals @ Formal Brieskorn modules @ Geometric (a,b)-modules @ Frescos
@ Gauss-Manin system.
