Derivation Lie Algebras Of New K-Th Local Algebras Of Isolated Hypersurface Singularities

Let (formula presented) be an isolated hypersurface singularity with mult. f / D m. Let J_k(f) be the ideal generated by all k-th order partial derivative of f. For 1 ≤ k ≤ m -1, the new object L_k(V)is defined to be the Lie algebra of derivations of the new k-th local algebra M_k(V), where (formula presented) Its dimension is denoted as δ_k(V). This number δ_k(V) is a new numerical analytic invariant. We compute L3.V/ for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ₃(V). We also formulate a sharp upper estimate conjecture for the δ_k(V) of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: (formula presented) and verify it for low-dimensional fewnomial singularities.