Derivation Lie algebras of new k-th local algebras of isolated hypersurface singularities

Abstract

. Let ( V, 0) = { ( z 1 , · · · , z n ) ∈ C n : f ( z 1 , · · · , 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 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 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 3 ( V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ 3 ( 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: δ ( k +1) ( V ) < δ k ( V ) , k ≥ 1 and verify it for low-dimensional fewnomial singularities.