Shape of zero set of homogeneous polynomial

Question

Let $f$ be a homogeneous polynomial in $d$ variables of degree $n$ over the real numbers. What does its zero set $V(f)$ look like? Is it a "hypersurface"? Is it connected in the metric topology of $\mathbb{R}^d$?

Answer 1

It may be a "hypersurface", but it might not be. For example, for $f(x_1,\ldots,x_d) = x_1^2 + x_2^2 + \ldots + x_d^2$ the zero-set consists only of $(0,\ldots,0)$. It certainly is connected, because if ${\bf x} \in V(f)$ then so is $t{\bf x}$ for any real $t$, so you can get from ${\bf x}$ to any ${\bf y} \in V(f)$ via ${\bf 0}$ in two straight line segments.