On the dimension of a new class of derivation lie algebras associated to singularities

Let (V, 0) = {(z₁, · · ·, z_n ) ∈ Cⁿ: f (z₁, · · ·, z_n ) = 0} be an isolated hypersurface singularity with mult( f ) = m. Let J_k ( f ) be the ideal generated by all k-th order partial derivatives 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 M_k (V):= O_n /(( f ) + J₁ ( f ) + · · · + J_k ( f )). Its dimension is denoted as δ_k (V). This number δ_k (V) is a new numerical analytic invariant. In this article we compute L₄ (V) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ₄ (V). We also verify a sharp upper estimate conjecture for the δ₄ (V) for large class of singularities. Furthermore, we verify another inequality conjecture: δ_(k+1) (V) < δ_k (V), k = 3 for low-dimensional fewnomial singularities.