Spectrahedral shadows and completely positive maps on real closed fields

  • Manuel Bodirsky

    Technische Universität Dresden, Dresden, Germany
  • Mario Kummer

    Technische Universität Dresden, Dresden, Germany
  • Andreas Thom

    Technische Universität Dresden, Dresden, Germany
Spectrahedral shadows and completely positive maps on real closed fields cover
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Abstract

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application we prove that the cone of copositive matrices of size is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model-theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital -linear completely positive map on a real closed field extension of .

Cite this article

Manuel Bodirsky, Mario Kummer, Andreas Thom, Spectrahedral shadows and completely positive maps on real closed fields. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1509