Almost all subgeneric third-order Chow decompositions are identifiable
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913408065994752 |
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| author | Torrance, Douglas A. Vannieuwenhoven, Nick |
| author_facet | Torrance, Douglas A. Vannieuwenhoven, Nick |
| contents | For real and complex homogeneous cubic polyomials in $n+1$ variables, we prove that the Chow variety of products of linear forms is generically complex identifiable for all ranks up to the generic rank minus two. By integrating fundamental results of [Oeding, Hyperdeterminants of polynomials, Adv. Math., 2012], [Casarotti and Mella, From non defectivity to identifiability, J. Eur. Math. Soc., 2021], and [Torrance and Vannieuwenhoven, All secant varieties of the Chow variety are nondefective for cubics and quaternary forms, Trans. Amer. Math. Soc., 2021] the proof is reduced to only those cases in up to $103$ variables. These remaining cases are proved using the Hessian criterion for tangential weak defectivity from [Chiantini, Ottaviani, and Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM J. Matrix Anal. Appl., 2014]. We also establish that the smooth loci of the real and complex Chow varieties are immersed minimal submanifolds in their usual ambient spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_06980 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Almost all subgeneric third-order Chow decompositions are identifiable Torrance, Douglas A. Vannieuwenhoven, Nick Algebraic Geometry 14C20, 14N05, 14Q15, 14Q20, 15A69, 15A72 For real and complex homogeneous cubic polyomials in $n+1$ variables, we prove that the Chow variety of products of linear forms is generically complex identifiable for all ranks up to the generic rank minus two. By integrating fundamental results of [Oeding, Hyperdeterminants of polynomials, Adv. Math., 2012], [Casarotti and Mella, From non defectivity to identifiability, J. Eur. Math. Soc., 2021], and [Torrance and Vannieuwenhoven, All secant varieties of the Chow variety are nondefective for cubics and quaternary forms, Trans. Amer. Math. Soc., 2021] the proof is reduced to only those cases in up to $103$ variables. These remaining cases are proved using the Hessian criterion for tangential weak defectivity from [Chiantini, Ottaviani, and Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM J. Matrix Anal. Appl., 2014]. We also establish that the smooth loci of the real and complex Chow varieties are immersed minimal submanifolds in their usual ambient spaces. |
| title | Almost all subgeneric third-order Chow decompositions are identifiable |
| topic | Algebraic Geometry 14C20, 14N05, 14Q15, 14Q20, 15A69, 15A72 |
| url | https://arxiv.org/abs/2112.06980 |