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

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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. ***