A note on some fiber-integrals

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We remark that the study of a fiber-integral of the type F (s):= f =s (ω/df) (ω/df) either in the local case where ρ ≡ 1 around 0 is C ∞ 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 → D is a proper holomorphic function on a complex manifold X, smooth outside f = 0 with ρ ≡ 1 near f = 0, for given holomorphic (n+1)--forms ω and ω', that a better control on the asymptotic expansion of F when s → 0, is obtained by using the Bernstein polynomial of the "frescos" associated to f and ω and to f and ω' (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.