A note on some fiber-integrals

TL;DR

This paper investigates fiber-integrals associated with complex functions, demonstrating that analyzing Bernstein polynomials of specific Brieskorn modules ('frescos') yields better asymptotic control than using the full Gauss-Manin system, especially in local cases.

Contribution

It introduces the use of Bernstein polynomials of frescos to improve asymptotic analysis of fiber-integrals over traditional methods.

Findings

Better asymptotic control using fresco Bernstein polynomials.

Explicit evaluation of Bernstein polynomials for certain polynomials.

Application to local and global complex analytic settings.

Abstract

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.