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

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

Authors :
Edoardo Ballico
Alessandra Bernardi
University of Trento [Trento]
Geometry, algebra, algorithms (GALAAD)
Inria Sophia Antipolis - Méditerranée (CRISAM)
Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)
COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)
European Project: 252367,EC:FP7:PEOPLE,FP7-PEOPLE-2009-IEF,DECONSTRUCT(2010)
Edoardo Ballico
Alessandra Bernardi
Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Nice Sophia Antipolis (... - 2019) (UNS)
Source :
Proceedings of the American Mathematical Society, Proceedings of the American Mathematical Society, 2012, 140, 10, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Proceedings of the American Mathematical Society, American Mathematical Society, 2012, 140, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Ballico, Edoardo ; Bernardi, Alessandra (2011) Symmetric tensor rank with a tangent vector : a generic uniqueness theorem. [Preprint]
Publication Year :
2012
Publisher :
HAL CCSD, 2012.

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 $\mathbb {P}^m$.<br />Comment: 7 pages

Details

Language :
English
ISSN :
00029939 and 10886826
Database :
OpenAIRE
Journal :
Proceedings of the American Mathematical Society, Proceedings of the American Mathematical Society, 2012, 140, 10, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Proceedings of the American Mathematical Society, American Mathematical Society, 2012, 140, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Ballico, Edoardo ; Bernardi, Alessandra (2011) Symmetric tensor rank with a tangent vector : a generic uniqueness theorem. [Preprint]
Accession number :
edsair.doi.dedup.....9be6125bc923db2cae29a5470521e01f
Full Text :
https://doi.org/10.1090/S0002-9939-2012-11191-8⟩