Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$ only at $z = 0$. Let $D\_i \subset {\mathbb C}^N$ be the divisor defined by $\partial\_i F$. 


My question is: is there a natural condition on $F$ to guarantee the following (or can we prove the following under the nondegenerate condition):


For any $I \subsetneq \{1, \ldots, N\}$, $F$ doesn't vanish identically on (any component of) the intersection


$$\bigcap\_{i \in I} D\_i.$$



A condition on isolated singularity
Assume that $F$ is quasihomogeneous.


A condition on $F$ would be the following. 
For each $k$ write $F=x\_kG+H\_k(x\_1,\dots,x\_{k-1},x\_{k+1},\dots, x\_N)$. Then $H\_k$ is quasihomogeneous.
If for each $k$ the polynomial $H\_k$ has an isolated singularity in $\mathbb{C}^{N-1}$ then $F$ does not vanish identically on any irreducible component of $\cap\_{i\in I} D\_i$ for any $I$ under consideration.


However, if $H\_k$ has a nonisolated singularity in $\mathbb{C}^{N-1}$ then $F$ does vanish identically on an irreducible component $\cap\_{i \in \{1,\dots, N\}, i\neq k} D\_i$.


To prove the first statement, note that it suffices to prove it for any $I$ with $N-1$ elements. Let $I\_k$ be the subset where we leave out $k$.
Also note that by the Euler relation for quasihomogeneous polynomials we have that the partials of $F$ form a complete intersection.
Therefore any irreducible component of $C\_k:=\cap\_{i\in I\_k} D\_i$ has dimension one. If $F$ vanishes on an irreducible component of $C\_k$ then by the Euler relation for quasihomogeneous polynomials we find that $x\_k \partial\_k F$ is zero on this particular component. If $\partial\_k F$ would be zero then $F$ would have a nonisolated singularity, hence $x\_k=0$.


Now the partial of $F$ wrt to $x\_i$ for $i\neq k$, when restricted to $x\_k=0$, coincide with the partial of $H$ wrt to $x\_i$. Since $H$ has an isolated singularity we find that no component of $C\_k$ is contained in $x\_k=0$. This proves the first statement.


The second statement follows along the same lines.


