Symmetrization maps and minimal border rank Comon's conjecture
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916474034061312 |
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| author | Mańdziuk, Tomasz Ventura, Emanuele |
| author_facet | Mańdziuk, Tomasz Ventura, Emanuele |
| contents | One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor $F\in(\mathbb{C}^n)^{\otimes d}$ for $d\geq 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $(\mathbb{C}^n)^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\leq d+1$. Our technical tools are border apolarity and border varieties of sums of powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetrization maps and minimal border rank Comon's conjecture Mańdziuk, Tomasz Ventura, Emanuele Algebraic Geometry Computational Complexity Primary: 14N07, Secondary: 15A69, 14C05, 68Q17 One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor $F\in(\mathbb{C}^n)^{\otimes d}$ for $d\geq 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $(\mathbb{C}^n)^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\leq d+1$. Our technical tools are border apolarity and border varieties of sums of powers. |
| title | Symmetrization maps and minimal border rank Comon's conjecture |
| topic | Algebraic Geometry Computational Complexity Primary: 14N07, Secondary: 15A69, 14C05, 68Q17 |
| url | https://arxiv.org/abs/2411.05721 |