1.2 - D-modules: an overview towards effectivity

Summary

Abstract

This paper presents the D module approach for the “geometric” study of the solutions and “generalized” solutions of a system of differential operators with holomorphic coefficients. One associates to such a system the quotient ring D/I of the ring D by a left ideal I and studies the properties of the complexes Rhom(D/I, F) (see subsection 1.4).

If we have a finite representation of the coefficients, for instance when they are polynomials, many results developed below are effective. Also many constructions which look rather abstract can be mechanized, in particular through standard basis computations (see [B.M.] [C] [G]). However the complexity of these algorithms is too high and makes them intractable in practice. The result of [Gr] shows that the membership problem has a double exponential complexity. A more geometric theory with simple exponential complexity (like in commutative algebra) does not exist yet and should be developed …

The first part of the paper presents in detail several constructions in the one variable case. The second part provides an introduction to holonomic D modules and Bernstein polynomials, it relies on the paper of B. Malgrange “Motivations and introduction to the theory of D module” which is published in this volume. We give an algorithm and a new bound for the computation of Bernstein's polynomial in the case of an isolated singularity.