Algebraic differential equations of periods integrals

TL;DR

This paper introduces a method to compute Bernstein polynomials of 'fresco' differential equations associated with period integrals, offering better control and simpler computation than traditional methods, especially for certain polynomial cases.

Contribution

It presents a new approach to analyze period integrals using algebraic differential equations and provides an explicit algorithm for specific polynomial cases.

Findings

The 'fresco' Bernstein polynomial offers improved control over asymptotic expansions.

The method simplifies computations compared to full Brieskorn modules.

Explicit examples demonstrate the algorithm's effectiveness.

Abstract

We explain that in the study of the asymptotic expansion at the origin of a period integral like $\gamma$z $\omega$/df or of a hermitian period like f =s $\rho$.$\omega$/df $\land$ $\omega$ /df the computation of the Bernstein polynomial of the "fresco" (filtered differential equation) associated to the pair of germs (f, $\omega$) 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 $\in$ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when $\omega$ is a monomial holomorphic volume form. Several concrete examples are given.