Concise tensors of minimal border rank
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910576952737792 |
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| author | Jelisiejew, Joachim Landsberg, J. M. Pal, Arpan |
| author_facet | Jelisiejew, Joachim Landsberg, J. M. Pal, Arpan |
| contents | We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczyńska-Buczyński and results of Jelisiejew-Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_05713 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Concise tensors of minimal border rank Jelisiejew, Joachim Landsberg, J. M. Pal, Arpan Algebraic Geometry Computational Complexity Commutative Algebra 15A69, 14C05, 68Q15 We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczyńska-Buczyński and results of Jelisiejew-Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$. |
| title | Concise tensors of minimal border rank |
| topic | Algebraic Geometry Computational Complexity Commutative Algebra 15A69, 14C05, 68Q15 |
| url | https://arxiv.org/abs/2205.05713 |