Logarithmic vector fields and multiplication table

This is a review article on the Gauss-Manin system associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system. We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminant and the bifurcation set . As applications, we establish signature formulae for Euler characteristics of real hypersurfaces and real complete intersections by means of these fields.