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Detection of the torsion classes in the Brieskorn modules of homogeneous polynomials

Let f∈C[X1,…,Xn] be a homogeneous polynomial and B(f) be the corresponding Brieskorn module, which is the quotient of the polynomial ring by some specific C-vector space and it has a C[t]-module structure. The main results detect torsion classes in the Brieskorn module using explicit computations with differential forms. We compute the torsion of the Brieskorn module B(f) for two variables in case of non-isolated singularities and show that torsion order is at most 1. In addition, we find some interesting families in which B(f) is torsion free even in case of non-isolated singularities. We exhibit several examples to compute the monomial basis for B(f) and the construction of torsion elements for n>2.