Hilbert’s 14-th Problem and Cox Rings

Hilbert’s Tenth Problem over Rings of Integers of Number Fields

This is a term paper for Thomas Scanlon’s Math 229, Model Theory, at UC Berkeley, Fall 2006. The goal of this survey is to understand Bjorn Poonen’s theorem about elliptic curves and Hilbert’s Tenth problem over rings of integers of number fields and to record and exposit a few ideas which may be useful in an attack with Poonen’s theorem as a starting point.

The Analogue of Büchi’s Problem for Some Rings of Functions

Büchi’s problem is a number theoretical question, a positive answer to which would imply the following strengthening of the negative answer to Hilbert’s Tenth Problem: the positive existential theory of the rational integers in the language of addition and a predicate for the property ‘x is a square’ would be undecidable. Büchi’s problem remains open. In this article (A) We discuss Büchi’s prob...

Hilbert’s Tenth Problem over Rings of Number-theoretic Interest

As Easy as Q: Hilbert’s Tenth Problem for Subrings of the Rationals and Number Fields

Hilbert’s Tenth Problem over the rationals is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings R ⊆ Q having the property that Hilbert’s Tenth Problem for R, denoted HTP(R), is Turing equivalent to HTP(Q). We are able to put several additional constraints on the rings R that we construct. Given any co...

Hilbert’s Machine and the Axiom of Infinity

Hilbert’s machine is a supertask machine inspired by Hilbert’s Hotel whose functioning leads to a contradiction that compromises the Axiom of Infinity. 1. Hilbert’s Machine In the following conceptual discussion we will make use of a theoretical device that will be referred to as Hilbert’s machine, composed of the following elements (see Figure 1): (1) An infinite tape similar to those of Turin...