Classification and degenerations of small minimal border rank tensors via modules
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912254202478592 |
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| author | Jagiełła, Jakub Jelisiejew, Joachim |
| author_facet | Jagiełła, Jakub Jelisiejew, Joachim |
| contents | We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification and degenerations of small minimal border rank tensors via modules Jagiełła, Jakub Jelisiejew, Joachim Algebraic Geometry Computational Complexity Commutative Algebra 14N07, 68Q17, 13E10 We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$. |
| title | Classification and degenerations of small minimal border rank tensors via modules |
| topic | Algebraic Geometry Computational Complexity Commutative Algebra 14N07, 68Q17, 13E10 |
| url | https://arxiv.org/abs/2409.06025 |