Non-invertible quasihomogeneus singularities and their Landau-Ginzburg orbifolds

According to the classification of quasihomogeneus singularities, any polynomial f defining such singularity has a decomposition f = f_κ + fadd. The polynomial f_κ is of the certain form while fadd is only restricted by the condition that the singularity of f should be isolated. The polynomial fadd is zero if and only if f is invertible, and in the non-invertible case fadd is arbitrary complicated. In this paper we investigate all possible polynomials fadd for a given non-invertible f. For a given f_κ we introduce the specific small collection of monomials that build up fadd such that the polynomial f = f_κ + fadd defines an isolated quasihomogeneus singularity. If (f,Z/2Z) is Landau-Ginzburg orbifold with such non-invertible polynomial f, we provide the quasihomogeneus polynomial f̄ such that the orbifold equivalence (f,Z/2Z) ~ (f̄, \) holds. We also give the explicit isomorphism between the corresponding Frobenius algebras.