On the degree of varieties of sum of squares

Abstract

We study the problem of how many different sum of squares decompositions a general polynomial <i>f</i> with SOS-rank <i>k</i> admits. We show that there is a link between the variety SOS_<i>k</i>(<i>f</i>) of all SOS-decompositions of <i>f</i> and the orthogonal group O(<i>k</i>). We exploit this connection to obtain the dimension of SOS_<i>k</i>(<i>f</i>) and show that its degree is bounded from below by the degree of O (<i>k</i>). In particular, for <i>k</i> = 2 we show that SOS₂(<i>f</i>) 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 <i>k</i> and obtain the degree in the special case <i>k</i> = 2.