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 $U = \C^n \setminus V(f)$foranynon-zeropolynomial$f \in \Q[x_1, …, x_n]$;1.$H^k(U, \C_U)$,$\C_U$istheconstantsheafon$U$withstalk$\C$\mathrm{.2.}$H^k(U, \Vsc)$,$\Vsc$isalocallyconstantsheafofrank1on$U$. 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.