Contributions to the model theory of valued fields

本文的研究重点是值域的模型理论。这门学科的起点可以看作是A.Robinson在1950年左右的工作，他在那里展示了代数闭值域理论的模型完备性。另一个重要的结果是J.Ax和S.Kochen在20世纪60年代证明了所谓的Ax-Kochen定理，它确定了特征为零的Henselian场的一阶理论，并作为推论得到了Artin猜想的一个渐近版本。赋值也是数域中解释的关键工具，例如J.Koenigsmann用普适公式定义有理数域中的整数的定理；随后J.Robinson，Poonen等人的工作。我们在下面描述了本研究的初步技术目标。在传统的数理逻辑中，陈述要么是假的，要么是真的。我们也说它们的真值要么是0，要么是1。然而，由Chang-Keisler和许多其他人发展的连续逻辑的逻辑设置，其中语句可以在紧凑空间中取任意值，甚至可以在实线中取任意值。许多基本的分析结构，如希尔伯特空间和L^1-格，都可以通过这种方式进行模型理论分析。在处理值域时，可以考虑实值设置，即，值组是实数的子组的值域，或者是任意值组或更高等级的值域。前者被用于Berkovich空间理论，而Hrushovski-Loeser模型理论版本将价值群视为可定义的对象。全局值域的Ben-Yaacov-Hrushovski理论是基于连续逻辑的，因此该理论仅限于第一级估值。对于这一点，目前还没有一阶类似物。我们将尝试在局部环境中确定这一限制是否必要，或者更高秩赋值的理论是否与连续逻辑兼容。一个肯定的答案将为全球有价值的油田打开类似问题的大门。由于后一个问题是开放的，即使是更温和的局部问题以前也没有被研究过，很有可能进一步的研究将导致意想不到的方向变化。这个项目属于EPSRC数学科学研究领域的逻辑和代数，与数论和代数几何有不同的联系。目前没有任何公司或合作者参与其中。

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.