The Algebraic Boundary of the Sonc-Cone

In this article, we explore a connection between nonnegativity, the theory of A-discriminants, and tropical geometry. We show that the algebraic strata of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each strata is contained in a rational variety called a positive discriminant. As an application, we characterization generic support sets for which the sonc cone is equal to the sparse nonnegativity cone, and we give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.