Discriminants and nonnegative polynomials

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

► We study the cones of nonnegative multivariate polynomials. ► Their boundaries are described by discriminants. ► Generally they have no log-polynomial type barrier functions. ► A degree formula is given for discriminants of several polynomials. ► Applications are shown in representing nonnegative polynomials.