DERIVATIONS OF LOCAL k-TH HESSIAN ALGEBRAS OF SINGULARITIES

Abstract

In our previous work, we introduced a series of new derivation Lie algebras $L_k(V)$ associated to an isolated hypersurface singularity $(V,0)$. These are new analytic invariants of singularities. Here, we investigate $L_2(V)$ for fewnomial isolated singularities and obtain the formula of ${\lambda }_k(V)$ (i.e., the dimension of $L_k(V)$) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for $L_2(V)$. This is a continuation of our previous work (Math. Z. 298:3-4 (2021), 1813–1829). We proposed two new conjectures for ${\tau }_k(V)$ and ${\lambda }_k(V)$ and we prove these conjectures for a large class of singularities.