The cohomology groups of the restriction Mwhere Y is a non-singular subvariety of X and ⊗ L denotes the left derived functor (cf. [12]) of the tensor product.2. The cohomology groups Ext i D X (M, K[[x 1 , . . . , x n ]]) with coefficients in the formal power series solutions of M, which equal to those with coefficients in the convergent power series solutions if M is regular holonomic (cf. ).3. The tensor product M⊗ O X N and, more generally, the torsion groups T or i O X (M, N ), as left D X -modules.
The localizationx n ] is an arbitrary non-constant polynomial.
The (algebraic) local cohomology groupsIt was proved by Kashiwara [14] that these are all holonomic systems (the second one is a finite dimensional vector space) if so are M and N .Let us remark that if K = C and M is Fuchsian along Y in the sense of Laurent and Moteiro-Fernandes [22], which is the case if M is regular holonomic in the sense of [19], then there exists an isomorphismin the derived category of sheaves of C-vector spaces; here O an X and O an Y denotes the sheaves of holomorphic functions on X and on Y respectively, and RHom the right derived functor of Hom . Thus roughly speaking, M • Y corresponds to the system of partial differential equations which the solutions of M restricted to Y satisfy. Similarly, M ⊗ O X N corresponds to the system which the product of solutions of M and of N satisfies.As was observed by Galligo [11] and was developed by several authors (e.g. [7], [36], [37], [27], [28], [29], [1], [35]) the notion of Gröbner basis and the Buchberger algorithm [6]are essential in the algorithmic study of D-modules as well as in computational algebraic geometry (cf.[8], [9]). By using Gröbner bases for the Weyl algebra, we give algorithms for computing the objects listed above under some conditions on M and N , which are certainly satisfied if M and N are holonomic. These algorithms also apply to the analytic counterparts of these functors as long as the input D-module is defined algebraically.We first give an algorithm for the restriction (Algorithm 5.4) when Y is a linear subvariety of arbitrary codimension under the condition that M is specializable along Y , which is the case with an arbitrary holonomic D X -module M. Here M is specializable along Y by definition if and only if there exists a nonzero b-function, or the indicial polynomial of M along Y . We also give an algorithm to compute the b-function (Algorithm 4.6).Our method consists in computing a free resolution of M that is adapted to the so-called V -filtration associated with Y . Such a free resolution tensored with D Y →X :Then we use information on the integral roots of the b-function to truncate the complex and obtain a complex of finitely generated free D Y -modules. The first author gave in [31] an algorithm for the case where Y is of codimension one without using free resolution.This algorithm for the restriction also solves the other problems by virtue of some isomorphisms provided by the D-module theory, especially those described in [14]. See Algorithm 6...
