Discriminants and nonnegative polynomials

For a semialgebraic set K in Rn, let Pd(K) = {f ∈ R[x]≤d: f(u) ≥ 0 ∀u ∈ K} be the cone of polynomials in x ∈ Rn of degrees at most d that are nonnegative onK. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irreducible hypersurface defined by the discriminant ∆(f) of f. When K = {x ∈ Rn: g1(x) = · · · = gm(x) = 0} is a real algebraic variety, we show that ∂Pd(K) lies on the hypersurface defined by the discriminant ∆(f, g1,..., gm) of f, g1,..., gm. When K is a general semialgebraic set, we show that ∂Pd(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 Pd(K) does not have a barrier of type − logϕ(f) when ϕ(f) is required to be a polynomial, but such a barrier exits if ϕ(f) is allowed to be semialgebraic. Some illustrating examples are shown