2015/12/22 by Daniel Barlet, Barlet, Daniel
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.1512.07062
arxiv created 2015/12/22 · openalex publication_date 2015/12/22 · arxiv updated 2015/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \not≡ 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 \not≡ 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.