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Discriminants and Nonnegative Polynomials

Jiawang Nie (njw***at***math.ucsd.edu)

Abstract: 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.

Keywords: barrier, discriminants, nonnegativity, polynomials, hypersurface, resultants, semialgebraic sets, varieties

Category 1: Linear, Cone and Semidefinite Programming (Other )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: arXiv:1002.2230, http://www.math.ucsd.edu/~njw, Preprint, Mathematics Department, UCSD, 2010

Download: [PDF]

Entry Submitted: 02/11/2010
Entry Accepted: 02/11/2010
Entry Last Modified: 04/23/2010

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