Algorithmic computation of de Rham cohomology of complements of complex affine varieties

Let X = C-n. In this paper we present an algorithm that computes the de Rha m cohomology groups H-dR(i)(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 in Oaku and Takayama (1999 ), who considered the case where Y is a hypersurface, and our methods from Walther (1999) for the computation of local cohomology. We further extend t he algorithm to compute de Rham cohomology groups with supports H-dR,Z(i)(U , C) where again U is an arbitrary Zariski-open subset of X and Z is an arb itrary Zariski-closed subset of Li. Our main tool is a generalization of the restriction process from Oaku and Takayama (in press) to complexes of modules over the Weyl algebra. The rest riction rests on an existence theorem on V-d-strict resolutions of complexe s that we prove by means of an explicit construction via Cartan-Eilenberg r esolutions. All presented algorithms are based on Grobner basis computations in the Wey l algebra and the examples are carried out using the computer system Kan by Takayama (1999). (C) 2000 Academic Press.