Sums of Squares on Real Algebraic Curves
نویسنده
چکیده
Given an affine algebraic variety V over R with compact set V (R) of real points, and a non-negative polynomial function f ∈ R[V ] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in R[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 localglobal principle, compactness of V (R) 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
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