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

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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.