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

We present an algorithm for computing a holonomic system for a definite integral of a holonomic function 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 algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.Comment: corrected typos; Sections 5 and 6 were slightly revised with results unchange

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

We give an algorithm to compute the following cohomology groups on U = \C^n \setminus V(f) for any non-zero polynomial f \in \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, \Vsc), \Vsc 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 \setminus Z, \C) where $Z$ is a general algebraic set. Our algorithm is based on computations of Gr\"obner bases in the ring of differential operators with polynomial coefficients.Comment: 38 page

Polynomial and rational solutions of holonomic systems

The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k where k is a subfield of the complex numbers.Comment: 20 page