Discriminants and Nonnegative Polynomials

For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the irreducible hypersurface defined by the discriminant of a single polynomial. When K is a real algebraic variety, we show that P_d(K) lies on the hypersurface defined by the discriminant of several polynomials. When K is a general semialgebraic set, we show that P_d(K) lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically P_d(K) does not have a log-polynomial type barrier, but a log-semialgebraic type barrier exits. Some illustrating examples are shown.