Advertisement

Minkowski Sums and Hadamard Products of Algebraic Varieties

  • Netanel Friedenberg
  • Alessandro OnetoEmail author
  • Robert L. Williams
Chapter
Part of the Fields Institute Communications book series (FIC, volume 80)

Abstract

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically, we explore when these are varieties and examine their properties in terms of those of the original varieties. This project was inspired by Problem 5 on Surfaces in [13].

MSC 2010 codes:

14M99 14N05 14Q15 14R99 

Notes

Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The second author was supported by G S Magnuson Foundation from Kungliga Vetenskapsakademien (Sweden).

References

  1. 1.
    Cristiano Bocci, Enrico Carlini, and Joe Kileel: Hadamard products of linear spaces, J. Algebra 448 (2016) 595–617.Google Scholar
  2. 2.
    Cristiano Bocci, Gabriele Calussi, Giuliana Fatabbi, and Anna Lorenzini: On Hadamard products of linear varieties, J. Algebra Appl. 16(8) (2017) 1750155, 22 pp.Google Scholar
  3. 3.
    Cristiano Bocci and Enrico Carlini: Idempotent Hadamard powers of varieties, in preparation.Google Scholar
  4. 4.
    María Angélica Cueto, Jason Morton, and Bernd Sturmfels: Geometry of the restricted Boltzmann machine, in Algebraic methods in statistics and probability II, 135–153, Contemp. Math. 516, American Mathematical Society, Providence, RI, 2010.Google Scholar
  5. 5.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
  6. 6.
    Joe Harris: Algebraic geometry, a first course, Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992.Google Scholar
  7. 7.
    Roger A. Horn and Charles R. Johnson: Topics in matrix analysis, Cambridge University Press, Cambridge, 1994.Google Scholar
  8. 8.
    Ludovico Lami and Marcus Huber: Bipartite depolarizing maps, J. Math. Phys. 57 (2016) 092201, 19 pp.Google Scholar
  9. 9.
    Joseph M. Landsberg: Tensors: geometry and applications, Graduate Studies in Mathematics 128, American Mathematical Society, Providence, RI, 2012.Google Scholar
  10. 10.
    Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.Google Scholar
  11. 11.
    Guido Montúfar and Jason Morton: Dimension of marginals of Kronecker product models, SIAM J. Appl. Algebra Geom. 1 (2017) 126–151.Google Scholar
  12. 12.
    William A. Stein et al.: Sage Mathematics Software (Version 7.6), The Sage Development Team, 2016, www.sagemath.org. SageMath, Inc., SageMathCloud Online Computational Mathematics, 2016. Available at https://cloud.sagemath.com/.
  13. 13.
    Bernd Sturmfels: Fitness, Apprenticeship, and Polynomials, in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  14. 14.
    Alessandro Terracini: Sulle V k per cui la varietà degli s h (h + 1)-seganti ha dimensione minore dell’ordinario, Rend. Circ. Mat. Palermo 31 (1911) 392–396.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Netanel Friedenberg
    • 1
  • Alessandro Oneto
    • 2
    Email author
  • Robert L. Williams
    • 3
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.INRIA Sophia Antipolis MéditerranéeSophia AntipolisFrance
  3. 3.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA

Personalised recommendations