, Volume 245, Issue 4, pp 725-760
Date: 04 Sep 2003

Sums of squares on real algebraic curves

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Given an affine algebraic variety V over ℝ with real points V(ℝ) compact 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 local-global principle, compactness of V(ℝ) can be relaxed to virtual compactness. The irreducible curves on which every non-negative polynomial is a sum of squares are classified. 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 virtually compact.

Mathematics Subject Classification (1991):14P05, 11E25, 14H99, 14P10, 44A60.
Dedicated to Eberhard Becker on the occasion of his 60th birthday