Let X=Cn. In this paper we present an algorithm that computes the cup product structure for the de Rham cohomology ring HdR•(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 efficient than the algorithm presented in Walther (J. Symbolic Comput. 29 (2000) 795-839). All presented algorithms are based on Gröbner basis computations in the Weyl algebra. © 2001 Elsevier Science B.V.
Walther, U. (2001). Computing the cup product structure for complements of complex affine varieties. Journal of Pure and Applied Algebra, 164(1–2), 247–273. https://doi.org/10.1016/S0022-4049(00)00157-2