Identifiability beyond Kruskal's bound for symmetric tensors of degree 4
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 Sym$^4(\mathbb{C}^{n+1})$, i.e., quartic hypersurfaces in a projective space $\mathbb{P}^n$, that have a decomposition in 2n+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 CayleyBacharach 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, "CayleyBacharach schemes and their canonical modules", Trans. Amer. Math. Soc. 339:443452, 1993.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.04211
 Bibcode:
 2017arXiv171204211A
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 16 pages