Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

We prove a Borsuk--Ulam-type zero theorem for the Stiefel manifold $V_{n,k}$ equipped with a free action of the hyperoctahedral group $B_k=(\mathbb{Z}/2)^k\rtimes S_k$. The existence of a zero reduces to a nonvanishing condition for explicit polynomials in the truncated ring ${R}_{n,k}=\mathbb{F}_2[