Spectrahedral shadows and completely positive maps on real closed fields
Manuel Bodirsky
Technische Universität Dresden, Dresden, GermanyMario Kummer
Technische Universität Dresden, Dresden, GermanyAndreas Thom
Technische Universität Dresden, Dresden, Germany
![Spectrahedral shadows and completely positive maps on real closed fields cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserials%2Fcover-jems.png&w=3840&q=90)
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