Is the set of sum of square (SOS) polynomials dense (in a sense to precise) in the set of non negative polynomials of degree less than $d$?
I don't even know how to ask a well posed question... here I don't preceise the topology, should we look the Zariski topology or else ?
In fact, my idea would be to relax a constraint of the form $$ p(x)>0 $$ where $f$ is a polynomial by the constraint $$ p(x)= SOS(x) $$
maybe other notions than density is better ...
I asked myself the very same question and found the Theorem 4.1 in this reference. It seems that your intuition was correct.