On the degree of varieties of sum of squares

We study the problem of how many different sums of squares decompositions a general polynomial \(f\) with SOS-rank \(k\) admits. We show that there is a link between the variety \(\mathrm{SOS}_k(f)\) of all SOS-decompositions of \(f\) and the orthogonal group \(\mathrm{O}(k)\). We exploit this connection to obtain the dimension of \(\mathrm{SOS}_k(f)\) and show that its degree is bounded from below by the degree of \(\mathrm{O}(k)\). In particular, for \(k=2\) we show that \(\mathrm{SOS}_2(f)\) is isomorphic to \(\mathrm{O}(2)\) and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank \(k\) and obtain the degree in the special case \(k=2\).