Polyhedral faces in Gram spectrahedra of binary forms

We analyze both the facial structure of the Gram spectrahedron $\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{m}(f)$ and of the Hermitian Gram spectrahedron ${\mathcal{H}}^{+}(f)$ of a nonnegative binary form $f \in $\mathbb{R}$[x, y]_{2d}$\mathrm{.}Weshowthatif$F \subseteq $\mathcal{H}$^{\scriptscriptstyle+}(f)$isapolyhedralfaceofdimension$k$then$\binom{k+1}{2} \leq d$\mathrm{.}Conversely,forall$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 $d\ge (k+1{)}^2$ the (symmetric) Gram spectrahedron of a general positive binary form $f\in \mathbb{R}[x,y{]}_{2d}$ contains a polyhedral face $F$ with $(\mathrm{r}\mathrm{k}(F),\dim (F))=(2(k+1),k)$.