Computing the Cup Product Structure for Complements of Complex Affine Varieties

Let X = C n. In this paper we present an algorithm that computes the cup product structure for the de Rham cohomology ring H dR (U; C) where U is the complement of an arbitrary Zariski-closed set Y in X. Our method relies on the fact that Tor is a balanced functor, a property which we make algorithmic, as well as a technique to extract explicit representatives of cohomology classes in a restriction or integration complex. We also present an alternative approach to computing V-strict resolutions of complexes that is seemingly much more eecient than the algorithm presented in 16]. All presented algorithms are based on Grr obner basis computations in the Weyl algebra.