Inequality conjectures on derivations of local k -th Hessain algebras associated to isolated hypersurface singularities
2021
Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $$L_{k}(V)$$
associated to (V, 0). Its dimension is denoted as $$\lambda _{k}(V)$$
. The $$L_{k}(V)$$
is a generalization of the Yau algebra L(V) and $$L_{0}(V)=L(V)$$
. These numbers $$\lambda _{k}(V)$$
are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $$L_1(V)$$
for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $$\lambda _{1}(V)$$
. We also formulate a sharp upper estimate conjecture for the $$L_k(V)$$
of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.
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