This is really more of a remark. The zero set of a homogeneous polynomial $f(x_0,\ldots, x_n)$ over a field, is a hypersurface in the projective space $\mathbb{P}^n_k$. These objects have been studied in algebraic geometry, and surrounding areas, for a long time.

Suppose that $k= \mathbb{C}$. When $d=\deg f=2$, then up to a linear change of variables, $f= x_0^2+\ldots x_m^2$. In particular, there is only one nonsingular quadric in a given dimension. Its geometry can be understood, by observing that it is homogenous under the action of the special orthogonal group. When $d> 2$, the story becomes much complicated. Except for plane cubics (elliptic curves) these are never homogeneous. All of the nonsingular hypersurfaces with $d,n$ fixed are diffeomorphic, however, as algebraic varieties or complex manifolds, they can be nonisomorphic. In fact, there are "continuous families" or moduli spaces of isomorphism classes of hypersurfaces. I'll stop with that.

This is really more of a remark. The zero set of a homogeneous polynomial $f(x_0,\ldots, x_n)$ over a field, is a hypersurface in the projective space $\mathbb{P}^n_k$. These objects have been studied in algebraic geometry, and surrounding areas, for a long time.

Suppose that $k= \mathbb{C}$. When $d=\deg f=2$, then up to a linear change of variables, $f= x_0^2+\dots + x_m^2$. In particular, there is only one nonsingular quadric in a given dimension. Its geometry can be understood, by observing that it is homogenous under the action of the special orthogonal group. When $d> 2$, the story becomes much complicated. Except for plane cubics (elliptic curves) these are never homogeneous. All of the nonsingular hypersurfaces with $d,n$ fixed are diffeomorphic, however, as algebraic varieties or complex manifolds, they can be nonisomorphic. In fact, there are "continuous families" or moduli spaces of isomorphism classes of hypersurfaces. I'll stop with that.