Toshinori Oaku

Algorithms for D-modules - restriction, tensor product, localization, and local cohomology groups

Abstract We consider an algebraic D -module M on the affine space, i.e. a system of linear partial differential equations with polynomial coefficients. We give an algorithm for computing the cohomology groups of the restriction of M to a linear subvariety by using a free resolution of M adapted to the V -filtration. Our algorithm works, at least, if M is holonomic. As applications, we obtain algorithms for computing tensor product, localization, and algebraic local cohomology groups of holonomic systems.

Algorithms for the b-function and D-modules associated with a polynomial

Let f be an arbitrary polynomial of n variables defined over a field of characteristic zero. We present algorithms for computing the b-function (Bernstein-Sato polynomial) of f, the D-module (the system of linear partial dierential equations) for f s , [ [

An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

Abstract We give an algorithm to compute the following cohomology groups on U= C n ⧹V(f) for any non-zero polynomial f∈ Q [x 1 ,…,x n ] : 1. H k (U, C U ), C U is the constant sheaf on U with stalk C . 2. H k (U, V ), V is a locally constant sheaf of rank 1 on U. We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H k ( C n ⧹Z, C ) where Z is a general algebraic set. Our algorithm is based on computations of Grobner bases in the ring of differential operators with polynomial coefficients.

Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients

It is proved that for a system of linear partial differential equations with polynomial coefficients, the Gröbner basis in the Weyl algebra is sufficient for the computation of the characteristic variety. In particular, this yields a correct algorithm of computing the singular locus of a holonomic system with polynomial coefficients. The characteristic variety is defined analytically, i.e. by using the ring of power series, and it has not been obvious that it can be computed by purely algebraic procedure. Thus the algorithm of computing the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients can be readily implemented on a computer algebra system.

A Gro¨bner basis method for modules over rings of differential operators

Abstract We study modules over the ring D 0 of differential operators with power series coefficients. For D 0 -modules, we introduce a new notion of F -Grobner basis and present an algorithmic method to compute it. Our method is more algebraic than that of Castro (1986, 1987) which is based on the Weierstrass-Hironaka division theorem. The essential point of our method consists in using a filtration of D 0 introduced by Kashiwara (1983). This enables us to extend some of the algorithmic methods for rings of power series to D 0 -modules. As applications, we can compute, in some cases, the characteristic variety, and the dimension of the space of solutions, of a system of linear partial differential equations via F -Grobner bases. The relation to previously known methods is also stated.

A localization algorithm for D -modules

Abstract We present a method to compute the holonomic extension of a D -module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent D -modules which are holonomic on the complement of an affine variety.

Tangent cone algorithm for homogenized differential operators

We extend Moras tangent cone or the ecart division algorithm to a homogenized ring of differential operators. This allows us to compute standard bases of modules over the ring of analytic differential operators with respect to sufficiently general orderings which are needed in the D-module theory.

Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities

A holonomic function is a differentiable or generalized function which satisfies a holonomic system of linear partial or ordinary differential equations with polynomial coefficients. The main purpose of this paper is to present algorithms for computing a holonomic system for the definite integral of a holonomic function with parameters over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithms, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.

Minimal Free Resolutions of HomogenizedD-modules

Homogenizing a module over the ring of differential operators, we define the notion of a minimal free resolution that is adapted to a filtration. We show that one can apply a modification of the algorithm of La Scala and Stillman to compute such a free resolution. By dehomogenization, one gets a free resolution of the original module that is small enough to compute, e.g. its restriction and integration. We have implemented our algorithm in a computer algebra system Kan and give examples by using this implementation.

Algorithms for finding the structure of solutions of a system of linear partial differential equations

for an unknown function u, where PI,. . .,P, are linear partial differential operators with polynomial coefficients. Systems of differential equations for various hypergeometric functions of several variables are typical examples (see Example 1 below). The aim of this paper is to present algorithms for finding the structure of the space of solutions of such a system of differential equations. Our method consists of the following three steps: