On the degree of varieties of sum of squares

TL;DR

This paper investigates the geometric structure of the set of sum of squares decompositions of a polynomial, revealing connections to orthogonal groups and providing dimension and degree bounds for these varieties.

Contribution

It establishes a link between SOS-decomposition varieties and orthogonal groups, deriving their dimension and degree bounds, especially for the case k=2.

Findings

The variety of SOS-decompositions for a polynomial of SOS-rank k relates to the orthogonal group O(k).

For k=2, the SOS-decomposition variety is isomorphic to O(2), with exact degree and dimension calculations.

The paper computes the dimension of polynomials with SOS-rank k and determines the degree for the case k=2.

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 $k=2$.