Mathematical Sciences: Topics in the Model Theory of p-Adic Fields

对环的实谱的研究已经产生了丰富的结果，包括C. Delzell对希尔伯特第十七问题连续解的一个优雅的证明。主要研究者的目的是证明一个积分定有理函数的Kochen表示的类似结果。这需要找到一个合适的p进类比，来模拟预排序的真正概念。通过研究p进场的普遍理论，首席研究员和她的合作者Luc Belair认为可以找到一个成功的“预估值”的定义。进一步的目标是研究“精细”p进谱，其中添加了n次幂集合的结构。第二个课题也与p进域的n次幂集的结构有关。首席研究员和她的合作者D. Macpherson最近定义了“p极小性”的概念，并证明了具有这种性质的场必须是p基闭的。进一步的重要问题是证明解析函数对p-基闭域的展开式是p-极小的，以及研究p-极小结构中可定义函数的可微性。模型理论是研究由公理定义的数学对象。模型理论代数将理论应用于特定的代数对象，在这种情况下应用于p进域。p进域是一种存在距离概念的数字系统，其方式类似于实数系统，但其中素数p起着特殊的作用。本研究的第一个问题是关于p进域的某些函数的表示，它揭示了它们的一些基本性质；问题是这种表现是否以一种可预测的方式变化。第二个问题，也是理解p-adics本质的核心，是问在一阶逻辑中p-adics中可定义的集合的几何是否与相应的实数集合具有共同的重要特征。＊＊＊

9401328 Haskell The study of the real spectrum of a ring has had rich consequences, including an elegant proof by C. Delzell of a continuous solution to Hilbert's seventeenth problem. The principal investigator aims to prove an analogous result for the Kochen representation of an integral-definite rational function. This requires finding an appropriate p-adic analogue to the real notion of a pre-ordering. By studying the universal theory of p-adic fields, the principal investigator and her collaborator Luc Belair think that a successful definition of "pre-valuation" can be found. A further goal is to study the "fine" p-adic spectrum, in which the structure of the sets of nth powers is added. The second project is also related to the structure of nth power sets in a p-adic field. The principal investigator and her collaborator D. Macpherson have recently defined the notion of "P-minimality" and have shown that a field with this property must be p-adically closed. Important further questions are to show that expansions of p-adically closed fields by analytic functions are P-minimal, and to study the differentiability of definable functions in P-minimal structures. Model theory is concerned with the study of mathematical objects which are defined by axioms. Model-theoretic algebra applies the theory to particular algebraic objects, in this case to p-adic fields. P-adic fields are number systems in which a notion of distance exists, in a way similar to that for the real number system, but in which the prime integer p plays a special role. The first problem in this research concerns a representation of certain functions of p-adic fields which reveals some of their essential properties; the question is whether this representation varies in a predictable way. The second problem, also at the heart of understanding the nature of the p-adics, asks whether the geometry of sets definable in the p-adics in first order logic has important features in com mon with the corresponding sets of real numbers. ***