Non-invertible quasihomogeneus singularities and their Landau-Ginzburg orbifolds

Abstract: 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 $(f,\mathbf{Z}/2\mathbf{Z})$ is Landau-Ginzburg orbifold with such non-invertible polynomial $f$, we provide the quasihomogeneus polynomial $\bar{f}$ such that the orbifold equivalence $(f,\mathbf{Z}/2\mathbf{Z}) \sim (\bar{f}, {id})$ holds. We also give the explicit isomorphism between the corresponding Frobenius algebras.