Concise tensors of minimal border rank
| Autor: | Jelisiejew, Joachim, Landsberg, J. M., Pal, Arpan |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 Buczy\'{n}ska-Buczy\'{n}ski and results of Jelisiejew-\v{S}ivic 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 $C^5\otimes C^5\otimes C^5$. Comment: v3, corrected a typo in statement of Theorem 1.8 |
| Databáze: | arXiv |
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