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