Algebraic differential equations of period-integrals

Daniel Barlet

Journal of Singularities
volume 25 (2022), 54-77

Received: 16 December 2020. Received in revised form: 21 August 2021.

DOI: 10.5427/jsing.2022.25c

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Abstract:

We explain that in the study of the asymptotic expansion at the origin of a period-integral or of a hermitian period the computation of the Bernstein polynomial of the "fresco" (filtered differential equation) associated to the pair of germs of a holomorphic function with a holomorphic volume form 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 the polynomial f in (n+1) variables has (n+2) monomials and is not quasi-homogeneous, by giving an explicit simple algorithm to produce a multiple of this Bernstein polynomial in the case of a monomial holomorphic volume form. Several concrete examples are given.

2010 Mathematical Subject Classification:

32S25-32S40

Key words and phrases:

Period-integral, Hermitian period, Formal Brieskorn Module, Geometric (a,b)-module, Fresco, Bernstein polynomial

Author(s) information:

Daniel Barlet
Institut Elie Cartan UMR 7502
Université de Lorraine, CNRS
INRIA et Institut Universitaire de France
BP 239 - F - 54506 Vandoeuvre-lès-Nancy Cedex, France
email: daniel.barlet@univ-lorraine.fr