Algebraic differential equations of period-integrals

We explain that in the study of the asymptotic expansion at the origin of a period integral like ∫ γz ω/df or of a hermitian period like ∫ f=s ρ.ω/df ∧ ω′/df the computation of the Bernstein polynomial of the ”fresco” (filtered differential equation) associated to the pair of germs (f, ω) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f ∈ C[x0, . . . , xn] has n+ 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when ω is a monomial holomorphic volume form. Several concrete examples are given. AMS Classification. 32 S 2532 S 40