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

Abstract

Let $X_{m,d}\subset {\mathbb{P}}^N$, $N:=\left(\genfrac{}{}{0ex}{}{m+d}{m}\right)-1$, 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 $t\le 1+\lfloor \left(\genfrac{}{}{0ex}{}{m+d-2}{m}\right)/(m+1)\rfloor$, 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 ${\mathbb{P}}^m$.

7 pages

References in corpus (1)

• Computing symmetric rank for symmetric tensors

Cited by in corpus (2)

• On schemes evinced by generalized additive decompositions and their regularity
• Canonical forms of Order-$k$ ($k=2,3,4$) Symmetric Tensors of Format $3\times \cdots \times 3$ Over Prime Fields