Recursive Koszul flattenings of determinant and permanent tensors
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910877067771904 |
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| author | Han, Jong In Ju, Jeong-Hoon Kim, Yeongrak |
| author_facet | Han, Jong In Ju, Jeong-Hoon Kim, Yeongrak |
| contents | We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on $\mathbf{R} (\det_n)$ completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks $\mathbf{R} (\det_4) = 12$ and $\mathbf{R} (\operatorname{perm}_4) = 8$ over arbitrary field of characteristic $\neq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recursive Koszul flattenings of determinant and permanent tensors Han, Jong In Ju, Jeong-Hoon Kim, Yeongrak Commutative Algebra Algebraic Geometry 14N07, 15A15, 15A69 We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on $\mathbf{R} (\det_n)$ completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks $\mathbf{R} (\det_4) = 12$ and $\mathbf{R} (\operatorname{perm}_4) = 8$ over arbitrary field of characteristic $\neq 2$. |
| title | Recursive Koszul flattenings of determinant and permanent tensors |
| topic | Commutative Algebra Algebraic Geometry 14N07, 15A15, 15A69 |
| url | https://arxiv.org/abs/2503.12032 |