  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.
