Defining Integers

TL;DR

This paper surveys recent progress in extending Hilbert's Tenth Problem to rational numbers and rings of integers, highlighting key developments and open questions in the field.

Contribution

It provides a comprehensive overview of recent advances and challenges in defining integers within the context of number theory and logic.

Findings

Progress in understanding Diophantine definability over number fields

Identification of key open problems in the area

Connections between Hilbert's Tenth Problem and number theory

Abstract

We survey the recent developments in the area that grew out of attempts to extend an analog of Hilbert's Tenth Problem to the field of rational numbers and rings of integers of number fields. The paper is based on a plenary talk the author gave at the North American Annual ASL meeting at the University of Notre Dame in May of 2009.