Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

Let X = C. In this paper we present an algorithm that computes the de Rham cohomology groups H dR(U, C) where U is the complement of an arbitrary Zariski-closed set Y in X . Our algorithm is a merger of the algorithm given by T. Oaku and N. Takayama ([7]), who considered the case where Y is a hypersurface, and our methods from [9] for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with support H dR,Z(U, C) where again U is an arbitrary Zariski-open subset of X and Z is an arbitrary Zariski-closed subset of U . Our main tool is the generalization of the restriction process from [8] to complexes of modules over the Weyl algebra. All presented algorithms are based on Gröbner basis computations in the Weyl algebra.