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Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 509–531 | Cite as

The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables

  • Edoardo Ballico
Article
  • 67 Downloads

Abstract

We give the stratification by the symmetric tensor rank of all degree d ≥ 9 homogeneous polynomials with border rank 5 and which depend essentially on at least five variables, extending previous works (A. Bernardi, A. Gimigliano, M. Idà, E. Ballico) on lower border ranks. For the polynomials which depend on at least five variables, only five ranks are possible: 5, d + 3, 2d + 1, 3d − 1, 4d − 3, but each of the ranks 3d − 1 and 2d + 1 is achieved in two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme A associated with the polynomial. The polynomial f depends essentially on at least five variables if and only if A is linearly independent (in all cases, f essentially depends on exactly five variables). The polynomial has rank 4d − 3 (resp. 3d − 1, resp. 2d + 1, resp. d + 3, resp. 5) if A has 1 (resp. 2, resp. 3, resp. 4, resp. 5) connected component. The assumption d ≥ 9 guarantees that each polynomial has a uniquely determined associated scheme A. In each case, we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme A.

Keywords

Symmetric tensor rank Symmetric rank Border rank Cactus rank 

Mathematics Subject Classification (2010)

14N05 

Notes

Acknowledgments

We thank the referee for his useful comments. This study is partially supported by the MIUR and GNSAGA of INdAM.

References

  1. 1.
    Albera, L., Chevalier, P., Comon, P., Ferreol, A.: On the virtual array concept for higher order array processing. IEEE Trans. Sig. Proc. 53(4), 1254–1271 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ballico, E.: Subsets of the variety X P n evincing the X-rank of a point of P n. Houston J. Math. 42(3), 803–824 (2016)MathSciNetMATHGoogle Scholar
  3. 3.
    Ballico, E., Bernardi, A.: Decomposition of homogeneous polynomials with low rank. Math. Z. 271, 1141–1149 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ballico, E., Bernardi, A.: Stratification of the fourth secant variety of Veronese variety via the symmetric rank. Adv. Pure Appl. Math. 4(2), 215–250 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bernardi, A., Gimigliano, A., Idà, M.: Computing symmetric rank for symmetric tensors. J. Symb. Comput. 46(1), 34–53 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bernardi, A., Ranestad, K.: The cactus rank of cubic forms. J. Symb. Comput. 50, 291–297 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E.P.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11-12), 1851–1872 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buczyńska, W., Buczyński, J.: Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes. J. Algebraic Geom. 23, 63–90 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buczyński, J., Ginensky, A., Landsberg, J.M.: Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture. J. London Math. Soc. (2) 88, 1–24 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chevalier, P.: Optimal separation of independent narrow-band sources-concept and performance, Signal Processing, Elsevier, 73, 27–48 special issue on blind separation and deconvolution (1999)Google Scholar
  11. 11.
    Comas, G., Seiguer, M.: On the rank of a binary form. Found. Comp. Math. 11(1), 65–78 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Comon, P., Lacoume, J.-L.: Independent component analysis Higher Order Statistics, pp 29–38. Elsevier, Amsterdam, London (1992)Google Scholar
  13. 13.
    Comon, P., Golub, G.H., Lim, L. -H., Mourrain, B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. 30(3), 1254–1279 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Comon, P., Mourrain, B.: Decomposition of Quantics in Sums of Powers of Linear Forms. Signal Process. Elsevier 53, 2 (1996)CrossRefMATHGoogle Scholar
  15. 15.
    De Lathauwer, L., Castaing, J.: Tensor-based techniques for the blind separation of DS-CDMA signals. Signal Process. 87(2), 322–336 (2007)CrossRefMATHGoogle Scholar
  16. 16.
    Dogǎn, M. C., Mendel, J.M.: Applications of cumulants to array processing. I. aperture extension and array calibration. IEEE Trans. Sig. Proc. 43(5), 1200–1216 (1995)CrossRefGoogle Scholar
  17. 17.
    Eisenbud, D., Harris, J.: Finite projective schemes in linearly general position. J. Algebraic Geom. 1(1), 15–30 (1992)MathSciNetMATHGoogle Scholar
  18. 18.
    Ellia, P. h., Peskine, Ch.: Groupes de points de P 2: caractère et position uniforme Algebraic Geometry (L’ Aquila, 1988), 111–116, Lecture Notes in Math., p 1417. Springer, Berlin (1990)Google Scholar
  19. 19.
    Granger, M.: Géométrie des schémas de Hilbert ponctuels. Mém. Soc. Math. France (N.S.) 2e série 8, 1–84 (1983)MATHGoogle Scholar
  20. 20.
    Hartshorne, R.: Algebraic Geometry. Springer-Verlag, Berlin (1977)CrossRefMATHGoogle Scholar
  21. 21.
    Jelisiejew, J.: An upper bound for the Waring rank of a form. Arch. Math. 102, 329–336 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jiang, T., Sidiropoulos, N.D.: Kruskal’s permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models. IEEE Trans. Sig. Proc. 52(9), 2625–2636 (2004)CrossRefGoogle Scholar
  23. 23.
    Iarrobino, A., Kanev, V.: Power Sums, Gorenstein Algebras, and Determinantal Loci Lecture Notes in Mathematics, vol. 1721. Springer-Verlag, Berlin, Appendix C by Iarrobino and Steven L. Kleiman (1999)Google Scholar
  24. 24.
    Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, vol. 128. American Mathematical Society, Providence (2012)Google Scholar
  25. 25.
    Landsberg, J.M., Teitler, Z.: On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10(3), 339–366 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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(3), 1084–1127 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    McCullagh, P.: Tensor Methods in Statistics. Monographs on Statistics and Applied Probability. Chapman and Hall (1987)Google Scholar
  28. 28.
    Ranestad, K., Schereyer, F. -O.: On the rank of a symmetric form. J. Algebra 346, 340–342 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoTrentoItaly

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