On the degree of varieties of sum of squares

We study the problem of how many different sum 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. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).