Almost all subgeneric third-order Chow decompositions are identifiable

For real and complex homogeneous cubic polynomials 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 (Adv Math 231:1308–1326, 2012), Casarotti...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2022-12, Vol.201 (6), p.2891-2905
Main Authors: Torrance, Douglas A., Vannieuwenhoven, Nick
Format: Article
Language:English
Subjects:
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Polynomials
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Summary:For real and complex homogeneous cubic polynomials 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 (Adv Math 231:1308–1326, 2012), Casarotti and Mella (J Eur Math Soc, 2021) and Torrance and Vannieuwenhoven (Trans Am Math Soc 374:4815–4838, 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 et al. (SIAM J Matrix Anal Appl 35(4):1265–1287, 2014). We also establish that the smooth loci of the real and complex Chow varieties are immersed minimal submanifolds in their usual ambient spaces.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-022-01224-8