Let
$\left(V,0\right)=\left\{\left({z}_{1},\dots ,{z}_{n}\right)\in {\u2102}^{n}:f\left({z}_{1},\dots ,{z}_{n}\right)=0\right\}$ be an isolated
hypersurface singularity with
$mult\left(f\right)=m$.
Let
${J}_{k}\left(f\right)$ be the ideal
generated by all
$k$th order
partial derivative of
$f$.
For
$1\le k\le m1$, the
new object
${\mathcal{\mathcal{L}}}_{k}\left(V\right)$
is defined to be the Lie algebra of derivations of the new
$k$th local
algebra
${M}_{k}\left(V\right)$, where
${M}_{k}\left(V\right):={\mathcal{\mathcal{O}}}_{n}\u2215\left(f+{J}_{1}\left(f\right)+\cdots +{J}_{k}\left(f\right)\right)$. Its dimension is
denoted as
${\delta}_{k}\left(V\right)$. This number
${\delta}_{k}\left(V\right)$ is a new numerical analytic
invariant. We compute
${\mathcal{\mathcal{L}}}_{3}\left(V\right)$
for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas
of
${\delta}_{3}\left(V\right)$.
We also formulate a sharp upper estimate conjecture for the
${\delta}_{k}\left(V\right)$ of
weighted homogeneous isolated hypersurface singularities and verify this conjecture for
large class of singularities. Furthermore, we formulate another inequality conjecture:
${\delta}_{\left(k+1\right)}\left(V\right)<{\delta}_{k}\left(V\right)$,
$k\ge 1$
and verify it for lowdimensional fewnomial singularities.
