Consider a polynomial $p \in \mathbb C[x_0,\cdots, x_n]$ that is homogeneous under a weighted $\mathbb C^*$ action (we can assume for simplicity that the weights are coprime). I would ideally like to know
What is/are necessary and sufficient conditions on the monomials and coefficients of a polynomial for its singular locus to be the origin.
By singular locus, I mean the variety defined as the common vanishing locus of $p$ and its derivatives. I think this probably requires some sort of sufficiently generic choice of terms (both monomials and coefficients), but I don't know how to properly characterize it.
For example, in $\mathbb C[x,y]$, setting $p = x^3$ is homogeneous under any weighted action, but its singular locus is $\mathbb V(x)$. Similarly, $p = (x-y)^3$ is homogeneous under the diagonal action, but its singular locus is $\mathbb V(x-y)$. If I take $p = x^2 + y^2$, however, the singular locus is $\mathbb V(x,y)$.
From playing around with various polynomials, I suspect I am looking for something like "$p$ must have 'sufficiently many monomials with generic coefficents'."
Edit: Sorry I was not clear; I'm not beginning with a fixed polynomial. I want to characterize polynomials whose singular locus is the origin. I suspect there should be a condition that says "if you pick exponents with such and such properties, then a polynomial with those monomials and generic complex coefficients will have singular locus at the origin."
If I pick any non-zero coefficent $a$, the polynomial $p=a x^3$, the singular locus will always be $\mathbb V(x)$, but for arbitrary non-zero coefficents $a,b$, the polynomial $p=a x^2 + b y^2$, the singular locus will always be $\mathbb V(x,y)$. So a condition might be "if the newton polytope is codimension 1 or less, for generic coefficients the singular locus is the origin".