This is a followup to my previous poorly-worded question.

Consider a finite collection of points $S \subset \mathbb N^n$ lying in a hyperplane $H$. These points define exponents of a collection of monomials in $\mathbb C[x_1, \cdots, x_n]$, which may be combined with some choice of coefficients to produce a polynomial that is homogeneous under the $\mathbb C^*$ action with weights determined by $H$.

Are there conditions on $S$ and/or $H$ such that by choosing generic coefficients, the resulting polynomial $p$ has singular locus at the origin?

I am happy to assume that the sum of the entries of $\alpha \in S$ (the naive degree) is greater than two, if this is easier.

In case it is still not clear, three examples: #1 is no good, but #2,3 are allowed.

If I choose $\{(3,0)\}\subset \mathbb N^2$, then for any non-zero coefficient $a$, the polynomial $p=ax^3$ has singularities away from the origin in $\mathbb C^2$.

If I choose $\{(3,0), (0,3)\} \subset \mathbb N^2$, then for any choice of non-zero coefficients $a,b$, the polynomial $p = ax^3 + by^3$ will be singular only at the origin.

If I choose $\{(6,0,0), (0,3,0), (0,0,2), (1,1,1))\} \subset \mathbb N^3$, then for non-zero $a,b,c,d$, the polynomial $a x^6 + b y^3 + cz^2 + d x y z$ has singularities only at the origin.

Please note that I am asking about singularities of the **affine** hypersurface, not the (weighted) projective hypersurface (hence *conical* singularity).

affine hypersurfacehas such a singularity. I hope I made that clear this time in my question. $\endgroup$