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Mediterranean Journal of Mathematics

, Volume 10, Issue 2, pp 643–654 | Cite as

Sets Computing the Symmetric Tensor Rank

  • Edoardo Ballico
  • Luca Chiantini
Article

Abstract

Let \({\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}\) denote the degree d Veronese embedding of \({\mathbb{P}^{r}}\). For any \({P\, \in \, \mathbb{P}^{N}}\), the symmetric tensor rank sr(P) is the minimal cardinality of a set \({\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}\) spanning P. Let \({\mathcal{S}(P)}\) be the set of all \({A \subset \mathbb{P}^{r}}\) such that \({\nu_{d}(A)}\) computes sr(P). Here we classify all \({P \,\in\, \mathbb{P}^{n}}\) such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of \({\nu_{d}(\mathbb{P}^{r})}\) . For such tensors \({P\, \in\, \mathbb{P}^{N}}\), we prove that \({\mathcal{S}(P)}\) has no isolated points.

Mathematics Subject Classification (2010)

14N05 15A69 

Keywords

Symmetric tensor rank Veronese embedding 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità of TrentoPovo (TN)Italy
  2. 2.Dipartimento di Scienze Matematiche ed Informatiche ’R. Magari’Università di SienaSienaItaly

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