On the degree of varieties of sum of squares

Andrew Ferguson; Giorgio Ottaviani; Mohab Safey El Din; Ettore Teixeira Turatti

doi:10.1016/j.jpaa.2024.107638

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arXiv · 2206.07473

On the degree of varieties of sum of squares

Andrew Ferguson·Giorgio Ottaviani·Mohab Safey El Din·Ettore Teixeira Turatti

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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 $k=2$.

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Andrew Ferguson, Giorgio Ottaviani, Mohab Safey El Din, Ettore Teixeira Turatti. 2022-06-15. On the degree of varieties of sum of squares. https://doi.org/10.1016/j.jpaa.2024.107638

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