论文标题
抽象的VergleichsStellenSätze用于预定的半场和半场
Abstract Vergleichsstellensätze for preordered semifields and semirings I
论文作者
论文摘要
实际代数通常被认为是对田野和环上某些类型的预订的研究。其核心主题包括分离定理，称为stitivstellensätze。但是，有一个新生的真实代数子场，研究了预订的半次数和半场，这是由于概率，图理论和理论计算机科学等的应用而动机。在这里，我们通过为其开发许多基础结果来为这个子领域做出贡献，其中两个抽象的Vergleichsstellensätze是我们的主要定理。 我们的第一个Vergleichsstellensatz指出，每个半场预订都是其总扩展的交集。我们将其应用于我们的第二个主要结果，即某些非架构的预分排到的vergleichsstellensatz，其中热带元素的同构具有重要作用。我们展示了该结果如何恢复Strassen的现有Vergleichsstellensatz，以及（通过后者）Krivine-Kadison-dubois的经典potivstellensatz。
Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellensätze. However, there is a nascent subfield of real algebra which studies preordered semirings and semifields, which is motivated by applications to probability, graph theory and theoretical computer science, among others. Here, we contribute to this subfield by developing a number of foundational results for it, with two abstract Vergleichsstellensätze being our main theorems. Our first Vergleichsstellensatz states that every semifield preorder is the intersection of its total extensions. We apply this to derive our second main result, a Vergleichsstellensatz for certain non-Archimedean preordered semirings in which the homomorphisms to the tropical reals play an important role. We show how this result recovers the existing Vergleichsstellensatz of Strassen and (through the latter) the classical Positivstellensatz of Krivine--Kadison--Dubois.
