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

Abstract

Let (V,0)={(z1,…,zn)∈Cn:f(z1,…,zn)=0} be an isolated hypersurface singularity with mult(f)=m. Let Jk(f) be the ideal generated by all k-th order partial derivatives of f. For 1≤k≤m−1, the new object Lk(V) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V), where Mk(V):=On/((f)+J1(f)+…+Jk(f)). Its dimension is denoted as δk(V). This number δk(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)<δk(V),k=3 for low-dimensional fewnomial singularities.