On the degree of varieties of sum of squares
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911782228983808 |
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| author | Ferguson, Andrew Ottaviani, Giorgio Din, Mohab Safey El Turatti, Ettore Teixeira |
| author_facet | Ferguson, Andrew Ottaviani, Giorgio Din, Mohab Safey El Turatti, Ettore Teixeira |
| contents | 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 $k=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07473 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the degree of varieties of sum of squares Ferguson, Andrew Ottaviani, Giorgio Din, Mohab Safey El Turatti, Ettore Teixeira Algebraic Geometry 14N07, 14N05 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 $k=2$. |
| title | On the degree of varieties of sum of squares |
| topic | Algebraic Geometry 14N07, 14N05 |
| url | https://arxiv.org/abs/2206.07473 |