FRG: Collaborative Research: Definability and Computability over Arithmetically Significant Fields

这个合作项目致力于研究一种重要的数学语言和由这种语言描述的对象。 我们脑海中的语言是多项式方程的语言，与高中代数课上第一次遇到的多项式方程相同。 尽管它的基本性质，这种语言拥有巨大的复杂性和描述能力，并不总是很好地理解数学家。 大约50年前，人们证明，没有计算机程序可以确定这种语言中的某种陈述是否正确，即使所描述的对象是自然数的集合，每个人从小就很熟悉。 换句话说，没有计算机程序可以确定一个多变量的任意多项式方程是否有整数解。另一方面，如果有人问关于有理数的同样问题，他就面临着关于多项式方程的许多基本问题之一，而这些问题的答案是未知的。 理解和试图解决这个问题和其他相关的问题需要互动和输入的几个领域的数学，如逻辑，数论，代数几何和拓扑。 同时，为多项式方程语言的研究而提出的问题和方法，也为上述数学领域带来了新的成果和研究方向。该项目涉及研究生培训，并将开发一个在线协作平台。 本项目考虑了数论、代数几何、模型论、可计算性理论和赋值理论等算术重要领域的可定义性和可计算性问题。 研究问题是在所有这些数学领域的交叉点。 虽然所采用的方法在很大程度上（虽然大多数肯定不总是）代数或几何的性质，问题起源于逻辑。 更具体地说，主要研究人员打算研究可计算性和可定义性的一阶和/或存在语言的环数域，其环的整数和无限代数扩展。 另一组相关的问题涉及函数域和所有特征的环的可计算性和可定义性。 该领域的一些主要突出问题涉及希尔伯特第十问题到有理数和代数整数环领域的扩展，有理数的最大阿贝尔扩展的一阶理论的可判定性，全局和局部域上函数域中赋值环的可定义性，代数闭域，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This collaborative project is dedicated to the study of an important mathematical language and the objects that are described by this language. The language we have in mind is the language of polynomial equations, the same polynomial equations one first encounters in an Algebra class in high school. Despite its basic nature this language possesses enormous complexity and descriptive power which were not always well-understood by mathematicians. About fifty years ago it was proved that no computer program can determine whether a certain kind of statement in this language is true even when the objects being described are sets of natural numbers, quite familiar to everyone since childhood. In other words, no computer program can determine whether an arbitrary polynomial equation in several variables has a solution in integer numbers. On the other hand, if one asks the same question about rational numbers, one is confronted with one of the many basic questions concerning polynomial equations to which the answer is unknown. Understanding and trying to tackle this question and other related ones requires interaction of and input from several fields of Mathematics such as Logic, Number Theory, Algebraic Geometry, and Topology. At the same time, the questions and methods developed for the study of the language of polynomial equations lead to new results and research directions in the areas of Mathematics mentioned above. This project involves graduate student training and it will develop an online collaboration platform. This project considers several problems in definability and computability over arithmetically significant fields, that is fields of importance to Number Theory, Algebraic Geometry, Model Theory, Computability Theory and Valuation Theory. The research problems are at the intersection of all these areas of Mathematics. While the methods employed are for the most part (though most definitely not always) algebraic or geometric in nature, the questions originate in Logic. More specifically, the Principal Investigators intend to study computability and definability in the first-order and/or existential language of rings over number fields, their rings of integers and their infinite algebraic extensions. Another set of related problems concerns computability and definability over function fields and rings of all characteristics. Some of the main outstanding questions in the area concern extensions of Hilbert’s Tenth Problem to the field of rational numbers and rings of algebraic integers, decidability of the first-order theory of the largest abelian extension of rational numbers, definability of valuation rings in function fields over global and local fields, algebraically closed fields, and other classes of arithmetically significant base fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.