On the description of identifiable quartics

TL;DR

This paper investigates the geometric conditions for the identifiability of quartic forms in five variables, extending algebraic geometry tools to establish criteria and algorithms for unique tensor decompositions.

Contribution

It provides a complete geometric description and criteria for identifiability of quartic forms in five variables for ranks 9 and above, including an effective algorithm for rank 12.

Findings

Criteria for identifiability for ranks ≥ 9

Complete geometric description of quartic forms in 5 variables

Effective algorithm for rank 12 decomposition

Abstract

In this paper we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of $3$ variables to more general cases. In particular, we focus on forms of degree $4$ in $5$ variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks $\geq 9$, filling the gap between rank $\leq 8$, covered by Kruskal's criterion, and $15$, the rank of a general quartic in $5$ variables. For the case $r=12$, we construct an effective algorithm that guarantees that a given decomposition is unique.