Symmetric tensor rank with a tangent vector: a generic uniqueness theorem

TL;DR

This paper establishes a generic uniqueness theorem for symmetric tensor decompositions involving tangent vectors, showing that under certain conditions, the decomposition of a tensor into linear forms is unique.

Contribution

It proves a new generic uniqueness result for symmetric tensor rank involving tangent vectors, extending previous understanding of tensor decompositions.

Findings

Unique decomposition of tensors involving tangent vectors under specified conditions

Explicit form of the tensor decomposition with unique linear forms

Conditions on parameters m, d, and t for the theorem to hold

Abstract

Let $X_{m,d}\subset \mathbb {P}^N$, $N:= \binom{m+d}{m}-1$, be the order $d$ Veronese embedding of $\mathbb {P}^m$. Let $\tau (X_{m,d})\subset \mathbb {P}^N$, be the tangent developable of $X_{m,d}$. For each integer $t \ge 2$ let $\tau (X_{m,d},t)\subseteq \mathbb {P}^N$, be the joint of $\tau (X_{m,d})$ and $t-2$ copies of $X_{m,d}$. Here we prove that if $m\ge 2$, $d\ge 7$ and $t \le 1 + \lfloor \binom{m+d-2}{m}/(m+1)\rfloor$, then for a general $P\in \tau (X_{m,d},t)$ there are uniquely determined $P_1,...,P_{t-2}\in X_{m,d}$ and a unique tangent vector $\nu$ of $X_{m,d}$ such that $P$ is in the linear span of $\nu \cup \{P_1,...,P_{t-2}\}$, i.e. a degree $d$ linear form $f$ associated to $P$ may be written as $$f = L_{t-1}^{d-1}L_t + \sum_{i=1}^{t-2} L_i^d$$ with $L_i$, $1 \le i \le t$, uniquely determined (up to a constant) linear forms on $\mathbb {P}^m$.