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 $\ge 9$, filling the gap between rank $\le 8$, 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.

摘要

在本文中，我们研究了特定形式（对称张量）的可识别性，目标是将最近针对 3 个变量的情况的方法扩展到更一般的情况。特别是，我们关注 5 个变量中 4 次的形式。借助于经典代数几何的工具，如希尔伯特函数、联络程序和塞尔构造，给出了秩的完整几何描述和可辨识准则$\ge 9$, 填补等级之间的差距$\le \mathrm{8个}$, 由 Kruskal 准则涵盖，15，一般四次方程在 5 个变量中的等级。对于r = 12的情况 ，我们构建了一个有效的算法来保证给定的分解是唯一的。