The Degree of \(\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})\)

  • Madeline Brandt
  • Juliette Bruce
  • Taylor Brysiewicz
  • Robert Krone
  • Elina Robeva
Part of the Fields Institute Communications book series (FIC, volume 80)


We provide a closed formula for the degree of \(\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})\). In addition, we test symbolic and numerical techniques for computing the degree of \(\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})\). As an application of our results, we give a formula for the number of critical points of a low-rank semidefinite programming problem. Finally, we provide evidence for a conjecture regarding the real locus of \(\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})\).

MSC 2010 codes:

14L35 20G20 15N30 



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 authors are very grateful to Jan Draisma for his tremendous help with understanding the Kazarnovskij Formula and to Kristian Ranestad for many helpful discussions. The authors thank Anton Leykin for performing the computation of \(\mathop{\mathrm{SO}}\nolimits (7, \mathbb{C})\). The first three authors would also like to thank the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany for their hospitality where some of this article was completed. The motivation for computing the degree of the orthogonal group came from project that started by the fifth author at the suggestion of Benjamin Recht. The first author was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400, and the second author was partially supported by the NSF GRFP under Grant No. DGE-1256259 and the Wisconsin Alumni Research Foundation.


  1. 1.
    Martin Aigner and Günter Ziegler: Proofs from The Book, Fourth edition, Springer-Verlag, Berlin, 2010.Google Scholar
  2. 2.
    Hans-Christian Graf von Bothmer and Kristian Ranestad: A general formula for the algebraic degree in semidefinite programming, Bull. Lond. Math. Soc. 41 (2009) 193–197.Google Scholar
  3. 3.
    Stephen Boyd and Lieven Vandenberghe: Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization, in Communications, Computation, Control, and Signal Processing, 279–287, Springer Science+Business Media, New York, 1997.Google Scholar
  4. 4.
    Taylor Brysiewicz: Experimenting to find many real points on slices of \(\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})\),
  5. 5.
    Samuel Burer and Renato Monteiro: Local minima and convergence in low-rank semidefinite programming, Math. Program. Ser. A 103 (2005) 427–444.Google Scholar
  6. 6.
    Harm Derksen and Gregor Kemper: Computational invariant theory, Encyclopaedia of Mathematical Sciences 130, Springer, Heidelberg, 2015.Google Scholar
  7. 7.
    Timothy Duff, Cvetelina Hill, Anders Jensen, Kisun Lee, Anton Leykin, and Jeff Sommars: Solving polynomial systems via homotopy continuation and monodromy, arXiv:1609.08722 [math.AG].Google Scholar
  8. 8.
    William Fulton and Joe Harris: Representation theory, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.Google Scholar
  9. 9.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at
  10. 10.
    Ira Gessel and Gérard Viennot: Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985) 300–321.Google Scholar
  11. 11.
    Michel X. Goemans and David P. Williamson: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42 (1995) 1115–1145.Google Scholar
  12. 12.
    Jonathan D. Hauenstein and Frank Sottile: alphaCertified software for certifying numerical solutions to polynomial equations, available at
  13. 13.
    James E. Humphreys: Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990.Google Scholar
  14. 14.
    B. Ya. Kazarnovskiĭ: Newton polyhedra and Bézout’s formula for matrix functions of finite-dimensional representations, Functional Anal. Appl. 21 (1987) 319–321.Google Scholar
  15. 15.
    James S. Milne: Algebraic number theory, v3.06, 2014, available at
  16. 16.
    Jiawang Nie, Kristian Ranestad, and Bernd Sturmfels: The algebraic degree of semidefinite programming, Math. Program. Ser. A 122 (2010) 379–405.Google Scholar
  17. 17.
    Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler: Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM J. Numer. Anal. 40 (2002) 2026–2046.Google Scholar
  18. 18.
    Andrew J. Sommese and Charles W. Wampler: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.Google Scholar
  19. 19.
    Bernd Sturmfels: Fitness, apprenticeship, and polynomials, in Combinatorial Algebraic Geometry,1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Madeline Brandt
    • 1
  • Juliette Bruce
    • 2
  • Taylor Brysiewicz
    • 3
  • Robert Krone
    • 4
  • Elina Robeva
    • 5
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Department of Mathematics & StatisticsQueen’s UniversityKingstonCanada
  5. 5.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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