Discriminants of cyclic homogeneous inequalities of three variables

In this article, we study the structure of the cone of semidefinite forms. It is a closed semialgebraic set but usually is not basic closed semialgebraic set. A discriminant is a defining equation of an irreducible component of algebraic boundary of this cone. We calculate discriminants using new tools - characteristic variety and local cones. A characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions. Local cone is a subcone of the PSD cone which corresponds to a maximal ideal. This theory works well for a family of polynomials which are invariant under an action of a finite group. After we construct an abstract general theory, we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d. We calculate some discriminants for d=3, 4, 5 and 6, and we show that this theory derives many new results.