P\'eriodes \'evanescentes et $(a,b)$-modules monog\`enes

TL;DR

This paper introduces a simple algebraic structure called (a,b)-modules to analyze the asymptotic behavior of vanishing periods, simplifying computations of Bernstein polynomials and exponents in asymptotic expansions.

Contribution

It proposes using minimal regular (a,b)-modules generated by one element to study asymptotics, providing explicit methods for computing Bernstein polynomials and exponents.

Findings

Bernstein polynomial computation is simplified for (a,b)-modules.

Explicit examples demonstrate the method's effectiveness.

The approach avoids integral shifts in asymptotic exponents.

Abstract

In order to describe the asymptotic behaviour of a vanishing period in a one parameter family we introduce and use a very simple algebraic structure : regular geometric (a,b)-modules generated (as left $\A-$modules) by one element. The idea is to use not the full Brieskorn module associated to the Gauss-Manin connection but a minimal (regular) differential equation satisfied by the period integral we are interested in. We show that the Bernstein polynomial associated is quite simple to compute for such (a,b)-modules and give a precise description of the exponents which appears in the asymptotic expansion which avoids integral shifts. We show a couple of explicit computations in some classical (but not so easy) examples.