JournalsrlmVol. 29, No. 3pp. 465–485

Identifiability beyond Kruskal’s bound for symmetric tensors of degree 4

  • Elena Angelini

    Università di Siena, Italy
  • Luca Chiantini

    Università di Siena, Italy
  • Nick Vannieuwenhoven

    Katholieke Universiteit Leuven, Belgium
Identifiability beyond Kruskal’s bound for symmetric tensors of degree 4 cover

A subscription is required to access this article.

Abstract

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in Sym4(Cn+1)\mathrm{Sym}^4(\mathbb C^{n+1}), i.e., quartic hypersurfaces in a projective space Pn\mathbb P^n, that have a decomposition in 2n+12n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley–Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, Cayley–Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339:443–452, 1993.

Cite this article

Elena Angelini, Luca Chiantini, Nick Vannieuwenhoven, Identifiability beyond Kruskal’s bound for symmetric tensors of degree 4. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 3, pp. 465–485

DOI 10.4171/RLM/817