| The results presented in this Memoire are framed within the algebraic study of the D -modules, where D is the ring of differential linear operators with analytic coefficients. The genesis of the D -modules is the theory of algebraic analysis of M. Sato, who considers a system of partial differential equations as a module of finite presentation on D . The first part of the thesis has as a goal to make a comparison between the theory of the Grobner bases and the theory developed by M. Janet. The aim of the second part of the thesis is make a generalization of the results and of the techniques of M. Janet to other rings of linear differential operators. This generalization is different from that accomplished by J. Briancon, Ph. Maisonobe and F. Castro and they will be specially interesting in order to extend the theory of the Grobner fan to the rings of linear differential operators with coefficients in C[[x1, &cdots; , xn]][ x-11,&cdots;,x-1n ] and C{lcub}x1, &cdots; , xn{rcub}[ x-11,&cdots;,x-1n ], between other rings.; We will briefly comment the content of each chapter. In Chapter 1 we summarized a series of concepts and known notions that we will need throughout the work, such as: Division Theorem in k[x 1, &cdots; , xn], Dickson's Lemma, Hilbert Base Theorem, Grobner Bases and Buchberger's Algorithm. In Chapter 2 we will study, according to M. Janet, the monomial systems and concretely the monomial modules. Chapter 3 is devoted to the formal study of systems of equations of partial derivatives. We will begin by presenting the notion of a system in canonical form and the notion of a completely integrable system; we will characterize the completely integrable systems and we will prove that if a system of differential equations has coefficients in a field, under certain conditions, each completely integrable system is a Grobner base and conversely. In Chapter 4 we will demonstrate that each completely integrable system has only one solution, depending on certain initial conditions. We will end this chapter, by showing a new proof of the equality ExtmDM ,O = 0 with m ≥ 1, where M is a D -submodule of a free module associated to a completely integrable system and O is the ring of the convergent series in n variables. In the following chapters we will generalize the methods of Janet to differential systems not necessarily in canonical form, so, in Chapter 5 we will introduce the notion of Grobner delta-base for a left ideal in the algebra Weyl, An(k) where k is a field and we will study the relation between both concepts. To end the chapter we can deduce adapted algorithms for the membership problem, the elimination problem and the syzygies problem by using Grobner delta-bases (instead of Grobner bases) that could be in some cases with more complexity. We end the Memorie by studyng Grobner delta-bases for sub-modules of a free module, which is the aim of Chapter 6. |
