Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914876639674368 |
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| author | Bodirsky, Manuel Kummer, Mario Thom, Andreas |
| author_facet | Bodirsky, Manuel Kummer, Mario Thom, Andreas |
| contents | 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 of this result we prove that the cone of copositive matrices of size $n\geq5$ 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 $\mathbb{R}$-linear completely positive map $R\to R$ on a real closed field extension $R$ of $\mathbb{R}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_06312 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields Bodirsky, Manuel Kummer, Mario Thom, Andreas Rings and Algebras Logic Operator Algebras Optimization and Control 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 of this result we prove that the cone of copositive matrices of size $n\geq5$ 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 $\mathbb{R}$-linear completely positive map $R\to R$ on a real closed field extension $R$ of $\mathbb{R}$. |
| title | Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields |
| topic | Rings and Algebras Logic Operator Algebras Optimization and Control |
| url | https://arxiv.org/abs/2206.06312 |