Discriminants and Nonnegative Polynomials

2010/02/10 by Nie, Jiawang · 2 citations
#14P05 #14P10 #14Q10 #90C25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1002.2230

Abstract

For a semialgebraic set K in Rⁿ, let P_d(K) be the cone of polynomials in Rⁿ of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=Rⁿ 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.

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