On a geometric method for the identifiability of forms

TL;DR

This paper presents a new geometric criterion for determining the uniqueness of tensor decompositions, leveraging Hilbert functions and Terracini's Lemma, offering a faster alternative to existing methods especially in non-general position cases.

Contribution

The paper introduces a novel geometric criterion for tensor identifiability based on Hilbert functions and Terracini's Lemma, improving computational efficiency over previous methods.

Findings

The criterion effectively tests decomposition uniqueness for ternary forms.

It operates within the same range as the reshaped Kruskal's criterion.

The proposed algorithm is faster, especially for non-general position point sets.

Abstract

We introduce a new criterion which tests if a given decomposition of a given ternary form $T$ of even degree is unique. The criterion is based on the analysis of the Hilbert function of the projective set of points $Z$ associated to the decomposition, and on the Terracini's Lemma which describes tangent spaces to secant varieties. The criterion works in a range for the length of the decomposition which is equivalent to the range in which the reshaped Kruskal's criterion (see [1]) works. Our criterion determines an algorithm for the identifiability of $T$ which is sensibly faster than algorithms based on the reshaped Kruskal's criterion, especially when the set of points $Z$ is not in general position.