  In this article, we explore the connections between nonnegativity, the theory
of $A$-discriminants, and tropical geometry. For an integral support set $A
\subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semi-algebraic
sets that are parametrized by families of tropical hypersurfaces. As an
application, we characterization generic support sets for which the sonc-cone
is equal to the sparse nonnegativity cone, and we describe a semi-algebraic
stratification of the boundary of the sonc-cone in the univariate case.
