Table of Contents
Geometry
Volume 2013, Article ID 614195, 3 pages
http://dx.doi.org/10.1155/2013/614195
Research Article

Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Department of Mathematics, University of Trento, Povo, 38123 Trento, Italy

Received 3 June 2013; Accepted 9 August 2013

Academic Editor: Anna Fino

Copyright © 2013 E. Ballico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Brachat, P. Comon, B. Mourrain, and E. Tsigaridas, “Symmetric tensor decomposition,” Linear Algebra and its Applications, vol. 433, no. 11-12, pp. 1851–1872, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Buczyński and J. M. Landsberg, “Ranks of tensors and a generalization of secant varieties,” Linear Algebra and its Applications, vol. 438, no. 2, pp. 668–689, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. Comon, G. Golub, L.-H. Lim, and B. Mourrain, “Symmetric tensors and symmetric tensor rank,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 3, pp. 1254–1279, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. M. Landsberg, Tensors: Geometry and Applications, vol. 128 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2012. View at MathSciNet
  5. J. M. Landsberg and Z. Teitler, “On the ranks and border ranks of symmetric tensors,” Foundations of Computational Mathematics, vol. 10, no. 3, pp. 339–366, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. de Silva and L.-H. Lim, “Tensor rank and the ill-posedness of the best low-rank approximation problem,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 3, pp. 1084–1127, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Kleppe, Representing a homogeneous polynomial as a sum of powers of linear forms [Thesis for the degree of Candidatum Scientiarum], Department of Mathematics, University of Oslo, 1999, http://folk.uio.no/johannkl/kleppe-master.pdf.
  8. G. Ottaviani, “An invariant regarding Waring's problem for cubic polynomials,” Nagoya Mathematical Journal, vol. 193, pp. 95–110, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. Ventura, “A note on the Waring ranks of reducible cubic forms,” http://arxiv.org/abs/1305.5394.
  10. A. Białynicki-Birula and A. Schinzel, “Representations of multivariate polynomials by sums of univariate polynomials in linear forms,” Colloquium Mathematicum, vol. 112, no. 2, pp. 201–233, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Białynicki-Birula and A. Schinzel, “Corrigendum to ‘Representatons of multivariate polynomials by sums of univariate polynomials in linear forms’,” Colloquium Mathematicum, vol. 125, no. 1, article 139, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. Ballico, “An upper bound for the symmetric tensor rank of a low degree polynomial in a large number of variables,” ISRN-Geometry, vol. 2013, Article ID 715907, 2 pages, 2013. View at Publisher · View at Google Scholar
  13. E. Ballico, “An upper bound for the tensor rank of an n-tensor,” submitted.
  14. E. Ballico, “An upper bound for the tensor rank,” ISRN Geometry, vol. 2013, Article ID 241835, 3 pages, 2013. View at Publisher · View at Google Scholar
  15. P. Comon and G. Ottaviani, “On the typical rank of real binary forms,” Linear and Multilinear Algebra, vol. 60, no. 6, pp. 657–667, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Iarrobino and V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci, vol. 1721 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1999, Appendix C by Iarrobino and Steven L. Kleiman. View at MathSciNet
  17. W. Buczyńska and J. Buczyński, “Secant varieties to high degree veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes,” http://arxiv.org/abs/1012.3563.
  18. J. Buczyński, A. Ginensky, and J. M. Landsberg, “Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture,” Journal of the London Mathematical Society, vol. 88, no. 2, pp. 1–24, 2013. View at Publisher · View at Google Scholar
  19. J. Jelisiejew, “An upper bound for the Waring rank of a form,” http://arxiv.org/abs/1305.6957. View at Google Scholar
  20. B. Ådlandsvik, “Joins and higher secant varieties,” Mathematica Scandinavica, vol. 61, no. 2, pp. 213–222, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. B. Ådlandsvik, “Varieties with an extremal number of degenerate higher secant varieties,” Journal für die Reine und Angewandte Mathematik, vol. 392, pp. 16–26, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Alexander and A. Hirschowitz, “La méthode d'Horace éclatée: application à l'interpolation en degré quatre,” Inventiones Mathematicae, vol. 107, no. 3, pp. 585–602, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. Alexander and A. Hirschowitz, “Polynomial interpolation in several variables,” Journal of Algebraic Geometry, vol. 4, no. 2, pp. 201–222, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. K. A. Chandler, “A brief proof of a maximal rank theorem for generic double points in projective space,” Transactions of the American Mathematical Society, vol. 353, no. 5, pp. 1907–1920, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. C. Brambilla and G. Ottaviani, “On the Alexander-Hirschowitz theorem,” Journal of Pure and Applied Algebra, vol. 212, no. 5, pp. 1229–1251, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. Bernardi, J. Brachat, and B. Mourrain, “A comparison of different notions of ranks of symmetric tensors,” http://arxiv.org/abs/1210.8169. View at Google Scholar
  27. A. Bernardi, P. Macias, and K. Ranestad, “Computing the cactus rank of a general form,” http://arxiv.org/abs/1211.7306. View at Google Scholar
  28. W. Buczyńska and J. Buczyński, “On the difference between the border rank and the smoothable rank of a polynomial,” http://arxiv.org/abs/1305.1726. View at Google Scholar
  29. E. Ballico and A. Bernardi, “Curvilinear schemes and maximum rank of forms,” http://arxiv.org/abs/1210.8171. View at Google Scholar