کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
401483 | 675369 | 2012 | 25 صفحه PDF | دانلود رایگان |
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.
Journal: Journal of Symbolic Computation - Volume 47, Issue 2, February 2012, Pages 167–191