The degree of $\text{SO}(n)$

TL;DR

This paper derives a closed formula for the degree of the special orthogonal group SO(n), introduces computational techniques for small n, and applies these results to semidefinite programming and conjectures about the real structure of SO(n).

Contribution

It provides the first explicit formula for the degree of SO(n) over characteristic zero fields and explores computational methods and applications in optimization and algebraic geometry.

Findings

Closed formula for the degree of SO(n)

Symbolic and numerical techniques for small n

Application to critical points in semidefinite programming

Abstract

We provide a closed formula for the degree of $\text{SO}(n)$ over an algebraically closed field of characteristic zero. In addition, we describe symbolic and numerical techniques which can also be used to compute the degree of $\text{SO}(n)$ for small values of $n$. As an application of our results, we give a formula for the number of critical points of a low-rank semidefinite programming optimization problem. Finally, we provide some evidence for a conjecture regarding the real locus of $\text{SO}(n)$.