  We provide a complete and explicit characterization of the real zeros of sums
of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative
polynomials independent of sums of squares. As a consequence, we derive an
exact determination of the number $B''_{n+1,2d}$ for all $n$ and $d$.
$B''_{n+1,2d}$ is defined to be the supremum of the number of zeros of all
homogeneous $n+1$-variate polynomials of degree $2d$ in the SONC cone. The
analogously defined numbers $B_{n+1,2d}$ and $B'_{n+1,2d}$ for the
nonnegativity cone and the cone of sums of squares were first introduced and
studied by Choi, Lam, and Reznick. In strong contrast to our case, the
determination of both $B_{n+1,2d}$ and $B'_{n+1,2d}$ for general $n$ and $d$ is
still an open question.
  Moreover, we initiate the study of the exposed faces of the SONC cone. In
particular, we explicitly consider small dimensions and analyze dimension
bounds on the exposed faces. When comparing the exposed faces of the SONC cone
with those of the nonnegativity cone we observe dimensional differences between
them.
