Generalized identifiability of sums of squares
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909303815798784 |
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| author | Ottaviani, Giorgio Turatti, Ettore Teixeira |
| author_facet | Ottaviani, Giorgio Turatti, Ettore Teixeira |
| contents | Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having $2$-rank equal to $1$ are exactly the squares. Such decompositions are never unique and they are divided into $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized identifiability of sums of squares Ottaviani, Giorgio Turatti, Ettore Teixeira Algebraic Geometry Commutative Algebra 14N07, 14N05, 11E25 Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having $2$-rank equal to $1$ are exactly the squares. Such decompositions are never unique and they are divided into $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms. |
| title | Generalized identifiability of sums of squares |
| topic | Algebraic Geometry Commutative Algebra 14N07, 14N05, 11E25 |
| url | https://arxiv.org/abs/2402.05189 |