论文标题
谱系阴影和实际封闭场上的完全正图
Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields
论文作者
论文摘要
在本文中，我们开发了新的方法，用于展示不是谱系阴影的凸半缘式集合。我们表征何时具有给定支持的一组非负多项式是在平方之和的方面是光谱的阴影。作为此结果的应用，我们证明了大小$ n \ geq5 $的共同矩阵的锥不是频谱的阴影，回答了Scheiderer的问题。我们的论点基于模型理论观察，即任何一个unital $ \ mathbb {r} $ - 在真实的封闭场扩展$ r $ r $ $ \ mathbb {r} $上定义了频谱黑色阴影的任何公式。
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 $\mathbb{R}$.
