Let $F \in \mathbb{Z}[x_0, \cdots, x_n]$ be a homogeneous polynomial. Let $V \subset \mathbb{P}^n(\mathbb{C})$ be a hypersurface (defined over $\mathbb{Q}$ say), given by a homogeneous polynomial $G(x_0, \cdots, x_n)$ say.

We say that $F$ *ramifies completely* on $V$ if there exists a positive integer $r > 1$ and polynomials $S,H$ such that $F(x_0, \cdots, x_n) = S^r + GH$. In other words, the image of $F$ with respect to the natural map $\mathbb{Z}[x_0, \cdots, x_n] \rightarrow \mathbb{Z}[x_0, \cdots, x_n)/I(V)$ is a perfect $r$-th power for some $r > 1$.

Unfortunately, assuming that $F$ is non-singular is not enough the exclude the possibility that $F$ ramifies on some hypersurface. Indeed, let $G(x_1, \cdots, x_n)$ be a non-singular polynomial of degree $d$ with respect to the variables $x_1, \cdots, x_n$, and put

$$\displaystyle F(x_0, \cdots, x_n) = x_0^d + G(x_1, \cdots, x_n).$$

It can easily be checked that $F$ is non-singular, and $F$ ramifies completely on the hypersurface given by $G = 0$.

Is there a way to classify those $F$ which ramifies completely on low degree hypersurfaces, namely hypersurfaces of degrees up to three?