JournalscmhVol. 87, No. 1pp. 113–140

Pure states, nonnegative polynomials and sums of squares

  • Sabine Burgdorf

    Universität Konstanz, Germany
  • Claus Scheiderer

    Universität Konstanz, Germany
  • Markus Schweighofer

    Universität Konstanz, Germany
Pure states, nonnegative polynomials and sums of squares cover
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Abstract

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial ff on a basic closed set KRnK\subset\mathbb{R}^n. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that KK has to be compact. This paper introduces the technique of pure states into commutative algebra. We show that this technique allows an approach to most of the recent archimedean Stellensätze that is considerably easier and more conceptual than the previous proofs. In particular, we reprove and strengthen some of the most important results from the last years. In addition, we establish several such results which are entirely new. They are the first that allow ff to have arbitrary, not necessarily discrete, zeros in KK.

Cite this article

Sabine Burgdorf, Claus Scheiderer, Markus Schweighofer, Pure states, nonnegative polynomials and sums of squares. Comment. Math. Helv. 87 (2012), no. 1, pp. 113–140

DOI 10.4171/CMH/250