On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

Let ( V , 0 ) = { ( z 1 , … , z n ) ∈ C n : f ( z 1 , … , z n ) = 0 } be an isolated hypersurface singularity with m u l t ( 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 ) + J 1 ( 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 L 4 ( V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ 4 ( V ) . We also verify a sharp upper estimate conjecture for the δ 4 ( 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.