In Schnell's note on Computing PicardFuchs Equations he gives a recursive method for computing residues on hypersurfaces. In short, if you have a meromorphic differential form $$ \frac{dw}{w^k}\wedge \rho $$ where $w$ is the local coordinate vanishing on a hypersurface $X$, you can write down a cohomologous meromorphic form $$ \frac{d\rho}{(k1)w^k} $$ He claims that you can rewrite this in a manner as above. He then also claims that once $k = 1$ you can write is down as $$ \frac{dw}{w}\wedge \alpha + \beta $$ and then the residue of this is $\alpha$. Unfortunately, I do not see an easy way to write down equations like this in general. Are there computer algebra tools which can accomplish this for me? The main test examples I am looking at are Fermat hypersurfaces (defined by $F=0$) and the differential forms $$ \frac{\Omega}{F} $$ where $$ \Omega = \sum_{i=0}^n(1)^i z_idz_0\wedge\cdots\wedge\hat{dz_i}\wedge\cdots\wedge dz_n $$
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This answer of mine might help. To compute in concrete examples the Jacobian ring any computer algebra tool (e.g. Macaulay2) will do.

$\begingroup$ Unfortunately, this is not what I am looking for, but, I appreciate your answer. I just want to get a better understanding of how to compute residues by hand just for the sake of it. $\endgroup$ – 54321user Jul 23 '17 at 18:58