Let $f\in \mathbb{R}[X_1,...,X_n]$ be a homogeneous polynomial of degree $d$. Let $F_{a}=\{(x_1,...,x_n)\in\mathbb{R}^n : f(x_1,...,x_n)=a\}$. For which $a$ $F_a$ is a submanifold in $\mathbb{R}^n$ ?
Edit: So, let $f=\sum_{i_1+...+i_n=d}A_{i_1...i_n}X_1^{i_1}...X_{n}^{i_n}$. Define $F:=f-a$. Then we should find all $a$ such that $\nabla F\neq0 $. We have that $\partial_{x_j}F=\sum_{i_1+...+i_n=d}A_{i_1...i_n}X_1^{i_1}...X_{j-1}^{i_{j-1}}X_{j}^{i_j -1}X_{j+1}^{i_{j+1}}...X_{n}^{i_n}$.
What is the simplest method to find zeros of $\nabla F$ ?