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 $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$.

Appendix: Macaulay2 Code

This section contains Macaulay2 [9] code for computing the degree of $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$. We typically compute the degree of $\mathop{\mathrm{O}}\nolimits (n, \mathbb{C})$, and divide by to 2 to obtain the degree of $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$, because this approach eliminates the polynomial of highest degree, the condition that the determinant equal 1.

First, we compute the degree of $\mathop{\mathrm{SO}}\nolimits (5)$ using Gröbner bases . The computation is done over the finite field $\mathbb{Z}/2\mathbb{Z}$ for $\mathop{\mathrm{O}}\nolimits (5, \mathbb{C})$ and the result is halved to give the degree of $\mathop{\mathrm{SO}}\nolimits (5, \mathbb{C})$.

deg1SO = n -> (     R := ZZ/2[x_(1,1)..x_(n,n)];     M := genericMatrix(R,n,n);     J := minors(1, M * transpose(M) - id_(R^n));     (degree J) // 2)

Our second function uses the package NumericalAlgebraicGeometry to solve the zero-dimensional system arising from a linear slice of the variety $\mathop{\mathrm{O}}\nolimits (3, \mathbb{C})$. The command solveSystem employs the standard method of polynomial homotopy continuation.

needsPackage ‘‘NumericalAlgebraicGeometry’’; deg2SO = n -> (     R := CC[x_(1,1)..x_(n,n)];     M := genericMatrix(R,n,n);     B := M * transpose(M) - id_(R^n);     polys := unique flatten entries B;     linearSlice := apply(binomial(n,2),         i -> random(1,R) - random(CC));     S := solveSystem(polys | linearSlice);     #S // 2)

We next provide code that computes the degree of $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$ using the package MonodromySolver. Again we do not include the determinant condition, but this time we do not need to halve the result. This is because our starting point, the identity matrix, lies on $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$ and this method only discovers points on the irreducible component corresponding to our starting point. The linear slices are parametrized by the t and c variables which are varied within the function monodromySolve to create monodromy loops. The method stops when ten consecutive loops provide no new points. Although it is possible that this stopping criterion is satisfied prematurely, in our case the program stopped at the correct number.

needsPackage ‘‘MonodromySolver’’; deg3SO = n -> (     d := binomial(n,2);     R := CC[c_1..c_d,         t_(1,1,1)..t_(d,n,n)][x_(1,1)..x_(n,n)];     M := genericMatrix(R,n,n);     B := M * transpose(M) - id_(R^n);     polys := unique flatten entries B;     linearSlice := for i from 1 to d list (         c_i + sum flatten for j from 1 to n list (     for k from 1 to N list t_(i,j,k)*x_(j,k)));     G := polySystem( polys | linearSlice);     setRandomSeed 0;     (p0, x0) := createSeedPair(G,         flatten entries id_(CC^n));     (V, npaths) = monodromySolve(G, p0, {x_0},         NumberOfNodes => 2, NumberOfEdges => 4);     # flatten points V.PartialSols)

Finally, we may use Theorem 1.1 to compute the degree of $\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$.

deg4SO = n -> (     r := n // 2;     M := matrix table(toList(1..r), toList(1..r),         (i,j) -> binomial(2*n-2*i-2*j, n-2*i));     2^(n-1) * det(M))