Combinatorial Algebraic Geometry pp 229-246 | Cite as

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

## Abstract

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## 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 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.

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