Collectanea Mathematica

, Volume 69, Issue 2, pp 205–220 | Cite as

The Hilbert function of some Hadamard products



In this paper we address the Hadamard product of not necessarily generic linear varieties, looking in particular at its Hilbert function. We find that the Hilbert function of the Hadamard product \(X\star Y\) of two varieties, with \(\dim (X), \dim (Y)\le 1\), is the product of the Hilbert functions of the original varieties X and Y. Moreover, the same result is obtained for generic linear varieties X and Y as a consequence of our showing that their Hadamard product is projectively equivalent to a Segre embedding.


  1. 1.
    Bocci, C., Calussi, G., Fatabbi, G., Lorenzini, A.: On Hadamard products of linear varieties. J. Algebra Appl. 16(8) (2017). doi: 10.1142/S0219498817501559
  2. 2.
    Bocci, C., Carlini, E., Kileel, J.: Hadamard products of linear spaces. J. Algebra 448, 595–617 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bocci, C., Carrucoli, D.: Gorenstein points via Hadamard products (in preparation)Google Scholar
  4. 4.
    Cueto, M.A., Morton, J., Sturmfels, B.: Geometry of the restricted Boltzmann machine. In: Viana, M., Wynn, H. (eds.) Algebraic Methods in Statistics and Probability. Contemporary Mathematics, vol. 516, pp. 135–153. American Mathematical Society (2010)Google Scholar
  5. 5.
    Cueto, M.A., Tobis, E.A., Yu, J.: An implicitization challenge for binary factor analysis. J. Symb. Comput. 45(12), 1296–1315 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Derksen, H., Sidman, J.: A sharp bound for the Castelnuovo Mumford regularity of subspace arrangements. Adv. Math. 172, 151–157 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Eisenbud, D.: The Geometry of Syzygies. Springer, Berlin (2005)MATHGoogle Scholar
  8. 8.
    Fatabbi, G., Harbourne, B., Lorenzini, A.: Inductively computable unions of fat linear subspaces. J. Pure Appl. Algebra 219, 5413–5425 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Franceschini, S., Lorenzini, A.: Fat points of \(\mathbb{P}^n\) whose support is contained in a linear proper subspace. J. Pure Appl. Algebra 160, 169–182 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Giuffrida, S., Maggioni, R., Ragusa, A.: On the postulation of \(0\)-dimensional subschemes on a smooth quadric. Pac. J. Math. 155, 251–282 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Horn, R.A., Mathias, R.: Block-matrix generalizations of Schur’s basic theorems on Hadamard products. Linear Algebra Appl. 172, 337–346 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Liu, S.: Inequalities involving Hadamard products of positive semidefinite matrices. J. Math. Anal. Appl. 243, 458–463 (2000b)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Liu, S.: On the Hadamard product of square roots of correlation matrices. Econometric Theory 18/19, 4 487 1007/703–704 (2002b/2003)Google Scholar
  14. 14.
    Liu, S., Neudecker, H.: Some statistical properties of Hadamard products of random matrices. Stat. Pap. 42, 475–487 (2001a)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, S., Neudecker, H., Polasek, W.: The Hadamard product and some of its applications in statistics. Statistics 26(4), 365–373 (1995a)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liu, S., Trenkler, G.: Hadamard, Khatri-Rao, Kronecker and other matrix products. Int. J. Inf. Syst. Sci. 4(1), 160–177 (2008)MathSciNetMATHGoogle Scholar
  17. 17.
  18. 18.
    Perrin, D.: Algebraic Geometry: An Introduction. Springer, Berlin (2007)Google Scholar

Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheUniversità di SienaSienaItaly
  2. 2.Dipartimento di Matematica e Informatica “Ulisse Dini”Università di FirenzeFlorenceItaly
  3. 3.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly

Personalised recommendations