Generic Gröbner Bases in D-modules and Application to Algebraic and Analytic Gröbner Fans

The contribution of this paper lies in two aspects. The first one deals with a natural notion of generic Gröbner (or standard) bases on an irreducible affine scheme for an ideal depending on parameters. This takes place in rings of differential operators and concerns the algebraic and the formal case. Thus we obtain a generalization of some known results in polynomial rings. The second aspect is an application of the first one to the study of the behaviour of the algebraic or analytic Gröbner fan of a monogeneous D-module depending on parameters. In substance, we prove that the Gröbner fan is generically constant. Introduction Fix an integer n ≥ 1. Let us denote by An(k) (resp. Dn and D̂n(k)) the ring of differential operators with coefficients in k[x] = k[x1, . . . , xn] (resp. in C{x} = C{x1, . . . , xn} and k[[x]] = k[[x1, . . . , xn]]) where k is a field of characteristic 1 0. In 2000 and 2001, A. Assi, F.J. Castro-Jiménez and M. Granger (see [ACG00] and [ACG01]) studied the notion of the (algebraic) Gröbner fan of a monogeneous An(k)module and of the analytic standard fan of a monogeneous Dn-module (we shall say “analytic Gröbner fan” in this paper) (see [MR88] for the commutative case, see also [St95]). The Gröbner fan is a combinatorial object which consists basically in taking into account the different filtrations of an An(k) (or D)-module and study the variations of the associated graded module. This is closely related to the notion of slopes of a D-module (see [ACG96]) and the notion of irregularity (see [La87] and [LM99]). To study the solutions of a regular holonomic system An(k)/I, the authors of [SST00] used the (algebraic) Gröbner fan associated with the ideal I ⊂ An(k). More recently, the author [Ba03b] gave a constructive and elementary proof of the existence of Bernstein-Sato polynomials for several analytic functions (see [Sa87a] and [Sa87b] for the original proof) by using the analytic Gröbner fan. In the present article, we shall focus on a natural question about Gröbner fans. Let I be an ideal depending on parameters, for example I ⊂ k[a1, . . . , a] ⊗k An(k) in the algebraic situation, or I ⊂ C{a1, . . . , a} ⊗C Dn in the analytic one. A natural question is to know how behaves the (algebraic) Gröbner fan associated with I|a0 ⊂ An(k) (where we specialize the parameters into a0) when a0 runs over k . We have a similar question about the behaviour of the analytic Gröbner fan associated with I|a0 ⊂ Dn when a0 runs over a small neighbourhood of 0 ∈ C. 1all the fields considered in this paper shall be of characteristic 0 2in this paper, ideal shall always mean left ideal 1