-

 

 

 




Optimization Online





 

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

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Programming Society