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 SOSₖ(f) of all SOS-decompositions of f and the orthogonal group O(k). We exploit this connection to obtain the dimension of SOSₖ(f) and show that its degree is bounded from below by the degree of O(k). In particular, for $k=2$ we show that SOS₂(f) is isomorphic to 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 $k=2$.