Almost all subgeneric third-order Chow decompositions are identifiable

Abstract

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.