When the Organizing Commitee of CADE began to choose the program of CADE-92, it was decided that D-modules would be a central topic at this conference.

The theory of D-modules is quite recent. It began in the late sixties and at first was considered to be quite abstract and difficult. Over the years the situation improved with the development of the theory and its applications. The organizers felt that it was time to try to introduce it to a larger audience interested in differential equations and computer algebra, since the theory of D-modules offers an excellent way to effectively handle linear systems of analytic PDEs.

Once this decision was made it was natural to ask Bernard Malgrange to be the “invité d'honneur” at CADE-92, with the task of lecturing about D-modules in a way adapted to an audience interested in effectivity. This was natural because Bernard Malgrange is not only one of the most famous mathematicians in this field, but also because he is perhaps the true originator of this direction. It is generally admitted that D-module theory began in the early seventies with the fundamental work of I. N. Berstein and of the Japanese school around M. Sato, but in fact Bernard Malgrange introduced the basic concepts ten years ago for the constant coefficients case (see his 1962 Bourbaki report “systèmes différentiels à coefficients constants”), and later for the general case (see his lectures at Orsay Cohomologie de Spencer (d'après Quillen)).

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.