These are possibly very basic questions. My knowledge in manifolds is quite limited and I would appreciate any comments!
Suppose I have $f \in \mathbb{Q}[x_1, ..., x_n]$ that is homogeneous of degree $d \geq 1$.
First Question: Does the zero locus of $f$ in $\mathbb{R}$ always form a real manifold?
Second question: When the zero locus over $\mathbb{R}$ does form a manifold, is the dimension always well defined and is it $n-1$? (or is it possibly not well defined)
Thank you very much!
If $0$ is a regular value of the map $\mathbb R^n \longrightarrow \mathbb R$ given by $(x_1,\dots,x_n) \mapsto f(x_1,\dots,x_n)$, then this is true by the submersion theorem. In general, however, this can fail: for instance, the zero locus of a homogeneous polynomial can be singular (e.g. consider the zero locus of $f(x,y)=x^2-y^2$ in the plane) and so cannot be a manifold.
Note that the regular value condition just translates to the statement that at every point such that $f(p)=0$, some partial derivative of $f$ does not vanish at $p$.
To answer your second question: in the case that $0$ is a regular value, this is in fact true. You can see this by looking at the proof of the submersion theorem. But there are some cases where $0$ is not a regular value of $(x_1,\dots,x_n) \mapsto f(x_1,\dots,x_n)$, but $\{f=0\}$ is still a manifold: for instance, consider $f(x,y)=x^2+y^2.$ In this case, the zero set is a point in the plane, and has dimension $0 = n-2$.