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Sums of squares on real algebraic curves

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Abstract.

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.

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Authors and Affiliations

  1. Institut für Mathematik, Fakultät 4, Universität Duisburg, 47048, Duisburg, Germany

    Claus Scheiderer

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  1. Claus Scheiderer
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Correspondence to Claus Scheiderer.

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Mathematics Subject Classification (1991):14P05, 11E25, 14H99, 14P10, 44A60.

Dedicated to Eberhard Becker on the occasion of his 60th birthday

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Scheiderer, C. Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003). https://doi.org/10.1007/s00209-003-0568-1

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