Sharp Lower Estimations for Invariants Associated with the Ideal of Antiderivatives of Singularities

Let $(V, 0)$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f: (\mathbb{C}^n, 0)\rightarrow (\mathbb{C}, 0)$. We introduce a new local algebra and a new derivation Lie algebra associated to $(V,0)$. These two new objects are defined by the ideal of antiderivatives with respect to the Tjurina ideal of $(V, 0)$. More precisely, let $I = (f, \frac{\partial f}{\partial x_1},\cdots, \frac{\partial f}{\partial x_n})$ and $\Delta(I) := \{g\mid g,\frac{\partial g}{\partial x_1},\cdots, \frac{\partial g}{\partial x_n}\in I\}$, then $A^\Delta(V) := \mathcal O_n/\Delta(I)$ and $L^\Delta(V) := \mathrm{Der}(A^\Delta(V),A^\Delta(V))$. Their dimensions as a complex vector space are denoted as $\beta(V)$ and $\delta(V)$ respectively. Both are new invariants of singularities. In this paper we study the new local algebra $A^\Delta(V)$ and the derivation Lie algebra $L^\Delta(V)$, and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimate conjectures for $\beta(V)$ and $\delta(V)$ when $(V, 0)$ are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities.