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Constructive Proofs of some Positivstellensätze for Compact Semialgebraic Subsets of ℝd

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Abstract

In a broad sense, positivstellensätze are results about representations of polynomials, strictly positive on a given set. We give proofs of some known positivstellensätze for compact semialgebraic subsets of ℝd, which are to a large extent constructive and elementary. The presented proofs extend and simplify arguments of Berr, Wörmann (Manuscripta Math. 104(2):135–143, 2001) and Schweighofer (J. Pure Appl. Algebra 166(3):307–319, 2002; SIAM J. Optim. 15(3):805–825, 2005).

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Notes

  1. Positivstellensatz (positivstellensätze, in plural) is a term from real algebraic geometry borrowed from German and meaning ‘positive-locus theorem’.

  2. This representation theorem is usually attributed to Kadison [5] and Dubois [6], though as pointed out by Marshall [7] and Prestel [8] it was proved before Dubois by Krivine [9, 10].

  3. Just use Farkas’ lemma, the implication (iv) ⇒ (iii) from Theorems 1 and 2 in [16].

  4. Handelman proves (H) under the assumption dim(S)=d (see [3, Theorem I.3]).

  5. This is easy to verify for various concrete choices of l 1,…,l k , e.g., in the case k=2d and {l 1≥0,…,l k ≥0}=[0,1]d. In the general situation, the boundedness of B follows from the fact that B has the same recession cone as {l 1≥0,…,l k ≥0}. See, for example, [23, Sect. 8.2].

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Acknowledgements

I would like to thank the anonymous referee for pointers to the literature and comments that helped to improve the presentation of the paper.

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  1. Institute for Mathematical Optimization, Faculty of Mathematics, University of Magdeburg, 39106, Magdeburg, Germany

    Gennadiy Averkov

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Communicated by Horst Martini.

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Averkov, G. Constructive Proofs of some Positivstellensätze for Compact Semialgebraic Subsets of ℝd . J Optim Theory Appl 158, 410–418 (2013). https://doi.org/10.1007/s10957-012-0261-9

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