Definability and decidability in global and local fields

粗略地说，如果存在一种算法，可以为每一个输入计算出正确的答案，那么这个数学问题就称为可判定的。数论中的可判定性问题由来已久，至少可以追溯到希尔伯特关于丢番图方程可解性的第十个问题：是否存在计算整数多项式是否有整数零的算法？目前，这是一个受到数论、算术几何、伽罗瓦理论、模型论(数理逻辑)和计算复杂性理论(计算机科学)影响的活跃的、高度交叉的研究领域。这里重要的开放问题是有理数的存在理论(即是否存在判定一个代数簇是否有有理点的算法)和其他全局领域的可判性，以及局部域的全一阶理论的正性特征的可判性。在这一领域，可判定性问题通常与可定义性问题密切相关：环或域的哪些子集是丢番图，即多项式的零集的投影，或者更一般地，可由环的语言中的一阶公式定义？本研究项目的目的是通过引入新的概念并以新颖的方式将所涉及的领域联系起来，对数论感兴趣的领域，特别是在全局域、局部域和代数域中的可定义性和可判定性问题做出贡献。它主要遵循申请人和他的合著者关于正特征的局部域的存在理论的可判定性和丢番图亨赛尔赋值环的结果，结合这一领域的最新突破，主要包括更深入地研究毕达哥拉斯数的p-进类似物，研究代数域的理论，某些代数域的可判定性，以及存在性域理论的可判定性。所采用的方法主要取自估值理论和模型理论代数，但带有一定的理论色彩。

A mathematical problem is called decidable if, roughly speaking there exists an algorithm that computes the correct answer for each input. Questions on decidability in number theory have a long history going back at least to Hilbert's Tenth Problem on solvability of diophantine equations: Does there exist an algorithm that computes whether an integer polynomial has an integer zero? Nowadays this is a lively and highly interdisciplinary research area with influences from number theory, arithmetic geometry, Galois theory, model theory (mathematical logic) and computational complexity theory (computer science).Important open questions here are the decidability of the existential theory of the rational numbers (i.e. does there exist an algorithm that decides whether an algebraic variety has a rational point) and other global fields, and the decidability of the full first-order theory of local fields in positive characteristic. Very often, decidability problems in this area are closely linked to questions of definability: Which subsets of a ring or field are diophantine, that is, projection of the zero set of a polynomial, or, more generally, definable by a first-order formula in the language of rings?The aim of this research project is to contribute to questions of definability and decidability in fields of number theoretic interest, in particular in global fields, local fields and algebraic fields, by introducing new ideas and connecting the involved areas in novel ways. It mainly follows the path laid out by the results of the applicant and his coauthors on the decidability of the existential theory of local fields in positive characteristic and on diophantine henselian valuation rings, incorporating the recent breakthroughs in this area.Goals include in particular a closer investigation of a p-adic analogue of the Pythagoras number, a study of the theory of algebraic fields, the decidability of certain algebraic fields, and decidability of existential theories of fields. The methods employed are taken mainly from valuation theory and model theoretic algebra, but with a number theoretic flavor.