Mathematische Zeitschrift

, Volume 271, Issue 3–4, pp 1141–1149 | Cite as

Decomposition of homogeneous polynomials with low rank

  • Edoardo BallicoEmail author
  • Alessandra Bernardi


Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of \({\mathbb{P}^m}\) into \({\mathbb{P}^{\tiny\left(\begin{array}{c}{\rm m+d} \\ {\rm d}\end{array}\right)-1}}\) but that its minimal decomposition as a sum of d-th powers of linear forms M 1, . . . , M r is \({F=M_1^d+\cdots + M_r^d}\) with r > s. We show that if s + r ≤ 2d + 1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.


Waring problem Polynomial decomposition Symmetric rank Symmetric tensors Veronese varieties Secant varieties 

Mathematics Subject Classification (2000)

15A21 15A69 14N15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander J., Hirschowitz A.: Polynomial interpolation in several variables. J. Algebraic Geom. 4(2), 201–222 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ballico, E., Bernardi, A.: Stratification of the fourth secant variety of Veronese variety via the symmetric rank. Math. A. G. ( Scholar
  3. 3.
    Bernardi A., Gimigliano A., Idà M.: Computing symmetric rank for symmetric tensors. J. Symb. Comput. 46, 34–55 (2011)zbMATHCrossRefGoogle Scholar
  4. 4.
    Brachat J., Comon P., Mourrain B., Tsigaridas E.P.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11–12), 1851–1872 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brambilla M.C., Ottaviani G.: On the Alexander-Hirschowitz theorem. J. Pure Appl. Algebra 212(5), 1229–1251 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Buczyński, J., Ginensky, A., Landsberg J.M.: Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture. Math. A. G. (arXiv:1007.0192)Google Scholar
  7. 7.
    Chiantini L., Ciliberto C.: Weakly defective varieties. Trans. Am. Math. Soc. 454(1), 151–178 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ciliberto C.: Geometric aspects of polynomial interpolation in more variables and of Waring’s problem. In: European Congress of Mathematics, I (Barcelona, 2000). Progr. Math., vol. 201, Birkhäuser, Basel, 289–316 (2001)Google Scholar
  9. 9.
    Ciliberto C., Mella M., Russo F.: Varieties with one apparent double point. J. Algebraic Geom. 13(3), 475–512 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ciliberto C., Russo F.: Varieties with minimal secant degree and linear systems of maximal dimension on surfaces. Adv. Math. 206(1), 1–50 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Comas G., Seiguer M.: On the rank of a binary form. Found. Comp. Math. 11(1), 65–78 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Comon P., Golub G.H., Lim L.-H., Mourrain B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. 30, 1254–1279 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Comon, P., Mourrain, B.: Decomposition of quantics in sums of powers of linear forms. In: Signal Processing, vol. 53. Elsevier, Amsterdam (1996)Google Scholar
  14. 14.
    Iarrobino, A., Kanev, V.: Power sums, Gorenstein algebras, and determinantal loci. In: Lecture notes in mathematics, vol. 1721. Springer, Berlin (1999) (Appendix C by Iarrobino and Steven L. Kleiman)Google Scholar
  15. 15.
    Lim L.-H., De Silva V.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. 30(3), 1084–1127 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Landsberg J.M., Teitler Z.: On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10(3), 339–366 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Mella M.: Singularities of linear systems and the Waring problem. Trans. Am. Math. Soc. 358(12), 5523–5538 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Mella M.: Base loci of linear systems and the Waring problem. Proc. Am. Math. Soc. 137(1), 91–98 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ranestad K., Schreyer F. O.: Varieties of sums of powers. J. Reine Angew. Math. 525, 147–181 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Vaughan, R.C., Wooley T.D.: Waring’s problem: a survey. In: Number theory for the millennium. III (Urbana, IL, 2000). A K Peters, Natick, pp. 301–340 (2002)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly
  2. 2.GALAAD, INRIA MéditerranéeSophia AntipolisFrance

Personalised recommendations