Real Algebraic and Analytic Geometry
Submission: 2002, September 21.
Given an affine algebraic variety V over IR with compact set V(IR) of real points, and a non-negative polynomial function f\in IR[V] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in IR[V]. We then specialize to the case where V is a curve. The notion of virtual compactness is introduced, and it is shown that in the local-global principle, compactness of V(IR) can be relaxed to virtual compactness. The irreducible curves are classified on which every non-negative polynomial is a sum of squares. All results are extended to the more general framework of preorders. Moreover, applications to the K-moment problem from analysis are given. In particular, Schmüdgen's solution of the K-moment problem for compact K is extended, for dim(K)=1, to the case when K is only virtually compact.
Mathematics Subject Classification (1991): 14P05,11E25,14H99,14P10,44A60.
Keywords and Phrases: Non-negative polynomials, sums of squares, positivity, preorders, archimedean, real algebraic curves, moment problem.
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