Suppose $f \in \mathbb R [x,y,z]$ is a homogeneous polynomial of degree $4$. Furthermore, suppose we can write
$$f(x,y,z)=\sum_i p_i(x,y,z)^2$$ where each $p_i$ is a homogeneous polynomial of degree $2$. My question:
Is there any "clean" intuition for the set of degree-$4$ $f$'s that can be written this way?
For example, if restriction $f$ to the unit sphere, do its maxima or minima have any relationships to one another? How do we distinguish this set from the full set of degree-$4$ homogeneous polynomials?
I know this question is vague. I'm trying to get an intuition for the "sum of squares" (SOS) condition in this case.