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Sums of squares on real algebraic curves

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2003

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http://nbn-resolving.de/urn:nbn:de:bsz:352-233026
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https://doi.org/10.1007/s00209-003-0568-1
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Mathematische Zeitschrift. 2003, 245(4), pp. 725-760. ISSN 0025-5874. eISSN 1432-1823. Available under: doi: 10.1007/s00209-003-0568-1

Zusammenfassung

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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510 Mathematik

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ISO 690SCHEIDERER, Claus, 2003. Sums of squares on real algebraic curves. In: Mathematische Zeitschrift. 2003, 245(4), pp. 725-760. ISSN 0025-5874. eISSN 1432-1823. Available under: doi: 10.1007/s00209-003-0568-1
BibTex
@article{Scheiderer2003squar-23302,
  year={2003},
  doi={10.1007/s00209-003-0568-1},
  title={Sums of squares on real algebraic curves},
  number={4},
  volume={245},
  issn={0025-5874},
  journal={Mathematische Zeitschrift},
  pages={725--760},
  author={Scheiderer, Claus}
}
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