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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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References

  1. 1.
    Ådlandsvik B.: Joins and higher secant varieties. Math. Scand. 61, 213–222 (1987)MathSciNetMATHGoogle Scholar
  2. 2.
    E. Ballico and A. Bernardi, Decomposition of homogeneous polynomials with low rank, arXiv:1003.5157v2 [math.AG] (to appear on Math. Z.).Google Scholar
  3. 3.
    E. Ballico and A. Bernardi, A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes, arXiv:1010.3546v2 [math.AG].Google Scholar
  4. 4.
    E. Ballico and L. Chiantini, A criterion for detecting the identifiability of symmetric tensors of size three, arXiv:1202.1741 [math.AG].Google Scholar
  5. 5.
    Bernardi A., Gimigliano A., Idà M.: Computing symmetric rank for symmetric tensors. J. Symbolic Comput. 46(no. 1), 34–53 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J. Brachat, P. Comon, B. Mourrain and E. P. Tsigaridas, Symmetric tensor decomposition, Linear Algebra Appl. 433 (2010), no. 11–12, 1851–1872.Google Scholar
  7. 7.
    W. Buczyńska and J. Buczyński, Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, arXiv:1012.3562v4 [math.AG].Google Scholar
  8. 8.
    J. Buczyński, A. Ginensky and J. M. Landsberg, Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture, arXiv:1007.0192v3 [math.AG].Google Scholar
  9. 9.
    J. Buczyński and J. M. Landsberg, Rank of tensors and a generalization of secant varieties, arXiv:0909.4262v4 [math.AG].Google Scholar
  10. 10.
    Chiantini L., Ciliberto C.: On the concept of k-secant order of a variety. J. London Math. Soc. (2) 73(no. 2), 436–454 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Comas G., Seiguer M.: On the rank of a binary form. Found. Comput. Math. 11(no. 1), 65–78 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    P. Comon, G. H. Golub, L.-H. Lim and B. Mourrain, Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl. 30 (2008) 1254–1279.Google Scholar
  13. 13.
    Couvreur A.: The dual minimum distance of arbitrary dimensional algebraicgeometric codes. J. Algebra 350(1), 84–107 (2012)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hartshorne R.: Algebraic Geometry. Springer-Verlag, Berlin (1977)MATHCrossRefGoogle Scholar
  15. 15.
    Kolda T., Bader B.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, Vol. 118, Amer. Math. Soc. Providence, 2012.Google Scholar
  17. 17.
    Landsberg J.M., Teitler Z.: On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10(no. 3), 339–366 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lim L.-H., de Silva V.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(no. 3), 1084–1127 (2008)MathSciNetCrossRefGoogle Scholar

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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