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用这种语言描述的对象。我们脑子里的语言是多项式方程的语言，和我们在高中代数课上第一次遇到的多项式方程是一样的。尽管它的基本性质，这种语言具有巨大的复杂性和描述能力，并不总是被数学家很好地理解。大约50年前，人们证明，没有计算机程序能够确定这种语言中的某种陈述是否正确，即使被描述的对象是一组自然数，这是每个人从小就非常熟悉的。换句话说，没有计算机程序可以确定一个包含多个变量的多项式方程是否有整数解。另一方面，如果一个人问同样的关于有理数的问题，他就会遇到许多关于多项式方程的基本问题之一，而这些问题的答案是未知的。理解和尝试解决这个问题和其他相关的问题需要从数学的几个领域，如逻辑学，数论，代数几何和拓扑学的相互作用和输入。同时，为研究多项式方程语言而提出的问题和方法在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。本课题研究了数论、代数几何、模型论、可计算论和估值论中重要的算术意义域的可定义性和可计算性的几个问题。研究的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本质上大部分是代数或几何的（尽管绝对不总是），但问题源于逻辑。更具体地说，主要研究者打算研究数域上环的一阶和/或存在语言的可计算性和可定义性，它们的整数环及其无限代数扩展。另一组相关问题涉及所有特征的函数域和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在有理数域和代数整数环上的推广，有理数的最大阿贝尔扩展的一阶理论的可裁断性，函数域上整体域和局部域上赋值环的可定义性，代数闭域，以及其他类型的算术有效基域。该奖项反映了美国国家科学基金会的法定使命，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.