An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation Original Research Article

We give an algorithm to compute the following cohomology groups on U=Cn-45 degree ruleV(f) for any non-zero polynomial fset membership, variantQ[x1,…,xn]: 1. Hk(U,CU), CU is the constant sheaf on U with stalk C. 2. image is a locally constant sheaf of rank 1 on 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 Hk(Cn-45 degree ruleZ,C) where Z is a general algebraic set. Our algorithm is based on computations of Gröbner bases in the ring of differential operators with polynomial coefficients.