On the degree of varieties of sum of squares

Abstract

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 $k=2$.

References in corpus (2)

• On generic and maximal k-ranks of binary forms
• Algebraic stories from one and from the other pockets

Cited by in corpus (1)

• Generalized identifiability of sums of squares