关于值域上半定可表集的性质
On semidefinite-representable sets over valued fields
AI总结：
本文研究了在值域上半定可表集及其优化问题的特性，提出基于Smith正则形的算法，并展示了非多面体的K-谱集及非K-谱集的半定可表集例子。
摘要：
实数上的多面体和谱集，或更一般地说它们在线性映射下的像，分别对应于线性和半定规划的可行集，构成了半定可表集的家族。本文研究了当数据取自值域K时这些集合及其相关优化问题的类比。对于K-多面体和K上的线性规划，我们提出基于Smith正则形计算的算法。我们证明了半定可表集的基本性质可扩展到值域设置。特别是，我们展示了非多面体的K-谱集以及在K上半定可表但非K-谱集的例子。
英文摘要：
Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.
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