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

Toshinori Oaku·Nobuki Takayama
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Abstract

We give an algorithm to compute the following cohomology groups on $U = \C^n \setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1. $H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U, \Vsc)$, $\Vsc$ 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 $H^k(\C^n \setminus Z, \C)$ 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.

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Toshinori Oaku, Nobuki Takayama. 1998-01-26. An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation. https://arxiv.org/abs/math/9801114

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