We analyze both the facial structure of the Gram spectrahedron $\mathrm{ Gram}(f)$ and of the Hermitian Gram spectrahedron $\mathcal{ H}^{ \scriptscriptstyle+}(f)$ of a nonnegative binary form $f \in \mathbb{ R}[x, y]_{ 2d}$. We show that if $F \subseteq \mathcal{ H}^{ \scriptscriptstyle+}(f)$ is a polyhedral face of dimension $k$ then $\binom{ k+1}{ 2} \leq d$. Conversely, for all $k \in \mathbb{ N}$ and $d \geq \binom{ k+1}{ 2}$ we show that the Hermitian Gram spectrahedron of a general positive binary form $f \in \mathbb{ R}[x, y]_{ 2d}$ with distinct roots contains a face $F$ which is a $k$-simplex and whose extreme points are rank-one tensors. For all $k \in \mathbb{ N}$ and $d \geq (k+1)^2$ the (symmetric) Gram spectrahedron of a general positive binary form $f \in \mathbb{ R}[x, y]_{ 2d}$ contains a polyhedral face $F$ with $(\mathrm{ rk}(F), \dim(F)) = (2(k+1), k)$.
Categories: math.OC math.AG