Contributions to the model theory of valued fields

本文主要研究赋值域的模型理论。这一课题的出发点可以看作是A.罗宾逊在1950年左右，在那里他表明了模型的完整性理论的代数闭值领域。另一个重要结果是由J. Ax和S. Kochen在20世纪60年代，确定了特征为零的Henselian域的一阶理论，并获得了Artin猜想的渐近版本作为推论。赋值也是解释数域的关键工具，例如J. Koenigsmann的定理通过一个普适公式定义了有理域中的整数;这是继J.罗宾逊、Poonen等人之后的工作。 我们在下面描述本研究的初始技术目标。传统上，在数理逻辑中，语句要么是假的，要么是真的。我们也说它们的真值是0或1。然而，有连续逻辑的逻辑设置，由Chang-Keisler和许多其他人开发，其中语句可以在紧凑的空间中取任意值，甚至在真实的行中。许多基本的分析结构，如希尔伯特空间和L^1格，都可以用这种方法进行模型论分析。当处理赋值域时，可以考虑真实的赋值设置，即具有值群的赋值域是真实的数的子群，或任意值群或更高秩。前者用于伯科维奇空间的理论，而Hrushovski-Loeser模型理论版本将值群视为可定义的对象。Ben-Yaacov - Hrushovski理论基于连续逻辑，因此仅限于秩1赋值。对于这一点，目前还没有一阶类似物。我们将试图在局部环境中确定这种限制是否必要，或者高阶赋值理论是否与连续逻辑相容。一个肯定的答案将为全球价值领域的类似问题开辟道路。由于后一个问题是完全开放的，甚至更温和的地方问题还没有研究过，这是很有可能的，进一步的调查将导致不可预见的变化的方向。这个项目福尔斯属于EPSRC数学科学研究领域的逻辑和代数，与各种连接到数论和代数几何。目前没有公司或合作者参与。

This research focuses on model theory of valued fields. The starting point for this subject can be seen as the work by A. Robinson around 1950, where he showed the model completeness of the theory of algebraically closed valued fields. Another important result is the so-called Ax-Kochen Theorem proved by J. Ax and S. Kochen in the 1960s, determining the first order theory of Henselian fields of characteristic zero, and obtaining an asymptotic version of a conjecture of Artin as a corollary. Valuations are also the key tool in interpretations in number fields, such as J. Koenigsmann's theorem defining the integers in the rational field by a universal formula; following work by J. Robinson, Poonen and others. We describe below an initial technical goal of the present research. Traditionally in mathematical logic, statements are either false or true. We also say that their truth value is either 0 or 1. There are however logical settings of continuous logic, developed by Chang-Keisler and many others, where statements can take arbitrary values in a compact space, or even in the real line. Many basic structures of analysis, such as Hilbert spaces, and L^1-lattices, become in this way accessible to model theoretic analysis. When working with valued fields, one can consider a real valued setting, i.e. a valued field with the value group being a subgroup of the real numbers, or an arbitrary value group or higher rank. The former is used in the theory of Berkovich spaces, whereas the Hrushovski-Loeser model-theoretic version treats the value group as a definable object. The Ben-Yaacov - Hrushovski theory of globally valued fields is based on continuous logic, which is therefore restricted to rank one valuations. For this there is no first-order analogue at the moment. We will attempt to determine in the local setting whether this restriction is necessary, or whether a theory of higher rank valuationsis compatible with continuous logic. A positive answer would open the way to similar questions for globally valued fields. As the latter question is wide open, and even the much more modest local question has not been studied before, it is quite possible that further investigation will lead to unforeseen changes of direction.This project falls within the EPSRC Mathematical sciences research areas of Logic and Algebra, with various connections to number theory and algebraic geometry. No companies or collaborators are currently involved.