Almost all subgeneric third-order Chow decompositions are identifiable

TL;DR

This paper proves that most third-order Chow decompositions of cubic polynomials are uniquely identifiable, extending previous results and using advanced algebraic geometry techniques to cover cases with up to 103 variables.

Contribution

It establishes generic complex identifiability for subgeneric ranks of cubic polynomials, reducing the problem to a finite number of cases and applying the Hessian criterion for proof.

Findings

Most subgeneric third-order Chow decompositions are identifiable.

The proof covers cases in up to 103 variables.

Smooth loci of Chow varieties are minimal submanifolds.

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.