Multidimensional local residues and . . .

Multidimensional local residues are fundamental objects in complex analysis and geometry. However, if the polar divisors of a meromorphic differential form are not in general position, the actual calculation of local residues is difficult in many cases. In this paper we study Grothendieck local residue from the viewpoint of $D$-modules. We mainly consider the case where the polar divisors are not in general position. We propose a new $\mathrm{a}\mathrm{p}\mathrm{p}\dot{\mathrm{r}}\mathrm{o}\mathrm{a}\mathrm{C}\mathrm{h}$ for calculating multidimensional local residues. In the appendix we consider the zero-dimensional transversal complete intersection case. We present a simple method for computing residues for this case. We use a computer algebra system Kan for Gr\"obner basis computation in Weyl algebra and a computer algebra system $\mathrm{R}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{r} $ for Gröbner basis computation, and primary decomposition in polynomial rings