Advertisement

Typical and admissible ranks over fields

Article
  • 45 Downloads

Abstract

Let \(X(\mathbb {R})\) be a geometrically connected variety defined over \(\mathbb {R}\) such that the set of all its complex points \(X(\mathbb {C})\) is non-degenerate. We introduce the notion of admissible rank of a point P with respect to X to be the minimal cardinality of a set \(S\subset X(\mathbb {C})\) of points such that S spans P and S is stable under conjugation. Any set evincing the admissible rank can be equipped with a label keeping track of the number of its complex and real points. We show that, in the case of generic identifiability, there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if X is a rational normal curve then there always exists a label for the generic element. We present two examples in which either the label doesn’t exist or the admissible rank is strictly bigger than the usual complex rank.

Keywords

Tensor rank Symmetric tensor rank Real symmetric tensor rank 

Mathematics Subject Classification

15A69 14N05 14P99 

Notes

Acknowledgements

We want to thank G. Ottaviani for very helpful and constructive remarks.

References

  1. 1.
    Angelini, E., Galuppi, F., Mella, M., Ottaviani, G.: On the number of waring decompositions for a generic polynomial vector. arXiv:1601.01869
  2. 2.
    Ballico, E., Chiantini, L.: Sets computing the symmetric tensor rank. Mediterr. J. Math. 10, 643–654 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bernardi, A., Blekherman, G., Ottaviani, G.: On real typical ranks. arXiv:1512.01853
  4. 4.
    Bernardi, A., Vanzo, D.: A new class of non-identifiable skew symmetric tensors. arXiv:1606.04158
  5. 5.
    Blekherman, G.: Typical real ranks of binary forms. Found. Comput. Math. 15, 793–798 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blekherman, G., Sinn, R.: Real rank with respect to varieties. Linear Algebra Appl. 505, 340–360 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Blekherman, G., Teitler, Z.: On maximum, typical and generic ranks. Math. Ann. 362, 1021–1031 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bocci, C., Chiantini, L.: On the identifiability of binary Segre products. J. Algebraic Geom. 22(1), 1–11 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bocci, C., Chiantini, L., Ottaviani, G.: Refined methods for the identifiability of tensors. Ann. Mat. Pura Appl. (4) 193(6), 1691–1702 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  11. 11.
    Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory. Grundlehren der Mathematischen Wissenschaften, vol. 315. Springer, Berlin (1997)MATHGoogle Scholar
  12. 12.
    Chiantini, L., Ottaviani, G.: On generic identifiability of 3-tensors of small rank. SIAM J. Matrix Anal. Appl. 33(3), 1018–1037 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Matrix Anal. Appl. 35(4), 1265–1287 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Comon, P., Ottaviani, G.: On the typical rank of real binary forms. Linear Multilinear Algebra 60(6), 657–667 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Galuppi, F., Mella, M.: Identifiability of homogeneous polynomials and Cremona Transformations. arXiv:1606.06895
  16. 16.
    Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)CrossRefMATHGoogle Scholar
  17. 17.
    Hauenstein, J.D., Oeding, L., Ottaviani, G., Sommese, A.J.: Homotopy techniques for tensor decomposition and perfect identifiability. arXiv:1501.00090
  18. 18.
    Hilbert, D.: Letter adressée à M. Hermite. Gesam. Abh. 2, 148–153 (1888)Google Scholar
  19. 19.
    Iarrobino, A., Kanev, V.: Power Sums, Gorenstein Algebras, and Determinantal Loci. Lecture Notes in Mathematics, vol. 1721. Springer, Berlin. Appendix C by Iarrobino and Steven L. Kleiman (1999)Google Scholar
  20. 20.
    Landsberg, J.M., Teitler, Z.: On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10(3), 339–366 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Palatini, F.: Sulla rappresentazione delle forme ternarie mediante la somma di potenze di forme lineari. Rom. Acc. L. Rend. 12, 378–384 (1903)MATHGoogle Scholar
  22. 22.
    Ranestad, K., Schreyer, F.: Varieties of sums of powers. J. Reine Angew. Math. 525, 147–181 (2000)MathSciNetMATHGoogle Scholar
  23. 23.
    Richmond, H.W.: On canonical forms. Quart. J. Pure Appl. Math. 33, 967–984 (1904)Google Scholar
  24. 24.
    Segre, C.: Sui complessi lineari di piani nello spazio a cinque dimensioni. Annali di Mat. pura ed applicata 27, 75–123 (1917)CrossRefMATHGoogle Scholar
  25. 25.
    Sylvester, J.J.: Collected Works. Cambridge University Press, Cambridge (1904)Google Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly

Personalised recommendations