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

Abstract

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 derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let $I = (f, \frac{\partial f}{\partial x_1},\ldots , \frac{\partial f}{\partial x_n})$ and $\Delta (I):= \{g\mid g,\frac{\partial g}{\partial x_1},\ldots , \frac{\partial g}{\partial x_n}\in I\}$, then $A^\Delta (V):= \mathcal O_n/\Delta (I)$ and $L^\Delta (V):= \textrm{Der}(A^\Delta (V),A^\Delta (V))$. Their dimensions as a complex vector space are denoted as $\beta (V)$ and $\delta (V)$, respectively. $\delta (V)$ is a new invariant 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 estimation 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. Lastly, we provide an application of $\beta (V)$ and $\delta (V)$ to distinguishing contact classes of singularities.