k -th Milnor numbers and k -th Tjurina numbers of weighted homogeneous singularities

Let (V, 0)⊂ (ℂ^n, 0) be a weighted homogeneous isolated hypersurface singularity. In this paper, we give explicit formulas of its k -th Milnor numbers and the k -th Tjurina numbers in terms of its weight type. Moreover, we propose a sharp lower bound conjecture for the k -th Tjurina numbers and verify this conjecture for binomial singularities. We also give a new characterization for the simple hypersurface singularities.