Existential Definability over Product Formula Fields

East Carolina University, Greenville NC

Investigators

Abstract

The main goal of this project is to increase our understanding of what is decidable and definable existentially in the language of rings. More specifically, we concentrate on issues of existential/Diophantine definability that have evolved from the solution of Hilbert's Tenth Problem. The main open problems in the area concern the existential decidability of rational numbers and rings of algebraic integers of number fields. New methods involving elliptic curves have shown promise in making these problems more approachable. We also investigate existential definability over function fields. In the case of function fields of characteristic 0 we also look at first-order definability problems since the corresponding existential definability problems seem out of reach at the moment. The interest in the questions of Diophantine definability and decidability dates back to the time of the solution of Hilbert's Tenth Problem (HTP). At the beginning of the XX century Hilbert asked the following question (among others): is there an algorithm that can determine whether an arbitrary polynomial equation in several variables and with integer coefficients has integer solutions? In the early 1970's, Yurii Matijasevich, building on the work by Martin Davis, Hilary Putnam and Julia Robinson showed that Diophantine sets and computably enumerable sets of integers were the same and thus showed that an algorithm sought by Hilbert did not exist. Matijasevich's result immediately raised another question which proved to be even more vexing: is there an algorithm as described above but for the solutions in rational numbers? This problem is unsolved to this day. As is often the case with difficult problems in Mathematics, HTP for rational numbers as well as its sister problem, HTP for the rings of integers of number fields, generated many new questions, quite interesting on their own, which the author of this proposal plans to investigate. Some of these questions turned out to be questions of Number Theory or Algebraic Geometry, but they in turn generated quite interesting consequences in Logic. The expectations are that questions originating in HTP will generate many new areas of interaction between Number Theory, Algebraic Geometry and Logic.