Finding residues of rational functions of two variables by cycles corresponding to cuts (Q1589064)

The author studies residues \[ \int_\Gamma f(z_1,z_2) dz_1 dz_2 \] of a rational function \(f(z_1,z_2)=\frac{P(z_1,z_2)}{Q(z_1,z_2)}\), \(P(z_1,z_2)\) and \(Q(z_1,z_2)\) being polynomials, where \(\Gamma\) is a two-dimensional singular cubic cycle not intersecting the singularities \(S=\{z=(z_1,z_2): Q(z_1,z_2)=0\}\). Some approach is developed which allows to calculate the residue if for the corresponding homology groups the expansions along some systems of generators are known.