Definability and decidability in global and local fields

全局和局部领域的可定义性和可判定性

• 批准号: 404427454
• 负责人: Professor Dr. Arno Fehm
• 金额: --
• 依托单位: Institut de Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/404427454?language=en
• 关键词: Definability; decidability; global; local; fields

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.

一个数学问题被称为可判定的，如果粗略地说，存在一个算法，计算每个输入的正确答案。数论中关于可判定性的问题有着悠久的历史，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：是否存在一种算法来计算整数多项式是否有整数零？目前这是一个活跃的、高度交叉的研究领域，受到数论、算术几何、伽罗瓦理论、模型论等学科的影响（数理逻辑）和计算复杂性理论（计算机科学）这里重要的未决问题是有理数存在理论的可判定性（即是否存在决定代数簇是否具有有理点的算法）和其他全局域，以及正特征局域场的完全一阶理论的可判定性。通常，该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，即多项式零集的投影，或者更一般地说，可由一阶定义环的语言公式？该研究项目的目的是通过引入新的想法并以新颖的方式连接所涉及的领域，为数论领域的可定义性和可判定性问题做出贡献，特别是在全球领域，局部领域和代数领域。它主要遵循申请人及其合著者关于正特征局部域的存在性理论的可判定性和关于丢番廷亨泽尔赋值环的结果所铺设的路径，并结合该领域的最新突破。目标特别包括对毕达哥拉斯数的p-adic模拟的更仔细的调查，对代数域理论的研究，某些代数域的可判定性，以及域的存在理论的可判定性。所采用的方法主要来自赋值理论和模型论代数，但具有数论的味道。