Definability and decidability in global and local fields

粗略地说，如果存在一种算法可以计算出每个输入的正确答案，那么一个数学问题就被称为可判定的。数论中关于可决性的问题有很长的历史，至少可以追溯到希尔伯特关于丢色图方程可解性的第十问题：是否存在一种算法来计算整数多项式是否有整数零？如今，这是一个充满活力和高度跨学科的研究领域，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）的影响。这里重要的开放性问题是有理数的存在论（即是否存在一种算法来决定一个代数变量是否有有理点）和其他全局域的可判定性，以及正特征局部域的全一阶理论的可判定性。通常，这一领域的可判定性问题与可定义性问题密切相关：环或域的哪些子集是丢番图的，即多项式零集的投影，或者更一般地说，可以用环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思想和以新颖的方式连接相关领域，为数论领域的可定义性和可判定性问题做出贡献，特别是在全局领域，局部领域和代数领域。它主要遵循申请人及其合著者关于积极特征的局部场的存在论的可决性和关于投番图henselian估值环的结果所铺设的路径，并结合了该领域的最新突破。目标特别包括对毕达哥拉斯数的p进模拟的更深入的研究，对代数场理论的研究，某些代数场的可判定性，以及场的存在理论的可判定性。所采用的方法主要来自估值理论和模型论代数，但又带有数论的味道。

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.