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In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of… More Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers.Less

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List price: $156.00 Copyright year: 2007 Publisher: Cambridge University Press Publication date: 11/9/2006 Binding: Hardcover Pages: 330 Size: 6.25" wide x 9.25" long x 0.75" tall Weight: 1.276 Language: English

AuthorTable of Contents

Acknowledgements

Introduction

In the beginning

Diophantine definitions and Diophantine sets

Diophantine classes: definitions and basic facts

Diophantine generation

Diophantine generation of integral closure and Dioph-regularity

Big picture: Diophantine family of a ring

Diophantine equivalence and Diophantine decidability

Weak presentations

Some properties of weak presentations

How many Diophantine classes are there?

Diophantine generation and Hilbert's Tenth Problem

Integrality at finitely many primes and divisibility of order at infinitely many primes

The main ideas

Integrality at finitely many primes in number fields

Integrality at finitely many primes over function fields

Divisibility of order at infinitely many primes over number fields

Divisibility of order at infinitely many primes over function fields

Bound equations for number fields and their consequences

Real embeddings

Using divisibility in the rings of algebraic integers

Using divisibility in bigger rings

Units of rings of W-integers of norm 1

What are the units of the rings of W-integers?

Norm equations of units

The Pell equation

Non-integral solutions of some unit norm equations

Diophantine classes over number fields

Vertical methods of Denef and Lipshitz

Integers of totally real number fields and fields with exactly one pair of non-real embeddings

Integers of extensions of degree 2 of totally real number fields

The main results for the rings of W-integers and an overview of the proof

The main vertical definability results for rings of W-integers in totally real number fields

Consequences for vertical definability over totally real fields

Horizontal definability for rings of W-integers of totally real number fields and Diophantine undecidability for these rings

Vertical definability results for rings of W-integers of the totally complex extensions of degree 2 of totally real number fields

Some consequences

Big picture for number fields revisited

Further results

Diophantine undecidability of function fields

Defining multiplication through localized divisibility

pth power equations over function fields I: Overview and preliminary results

pth power equations over function fields II: pth powers of a special element

pth power equations over function fields III: pth powers of arbitrary functions

Diophantine model of Z over function fields over finite fields of constants

Bounds for function fields

Height bounds

Using pth powers to bound the height

Diophantine classes over function fields

The weak vertical method revisited

Weak vertical method applied to non-constant cyclic extensions

The weak vertical method applied to constant field extensions

Vertical definability for large subrings of global function fields

Integrality at infinitely many primes over global function fields

The big picture for function fields revisited

Mazur's conjectures and their consequences

The two conjectures

A ring version of Mazur's first conjecture

First counterexamples

Consequences for Diophantine models

Results of Poonen

A statement of the main theorem and an overview of the proof

Properties of elliptic curves I: Factors of denominators of points

Properties of elliptic curves II: The density of set of "largest" primes

Properties of elliptic curves III: Finite sets looking big

Properties of elliptic curves IV: Consequences of a result of Vinograd

Construction of sets T[subscript 1](P) and T[subscript 2](P) and their properties

Proof of Poonen's theorem

Beyond global fields

Function fields of positive characteristic and of higher transcendence degree or over infinite fields of constants

Algebraic extensions of global fields of infinite degree

Function fields of characteristic 0

Recursion (computability) theory

Computable (recursive) functions

Recursively enumerable sets

Turing and partial degrees

Degrees of sets of indices, primes, and products

Recursive algebra

Recursive presentation of Q

Recursive presentation of other fields

Representing sets of primes and rings of S-integers in number fields

Number theory

Global fields, valuations, and rings of W-integers

Existence through approximation theorems

Linearly disjoint fields

Divisors, prime and composite, under extensions

Density of prime sets

Elliptic curves

Coordinate polynomials

Basic facts about local fields

Derivations

Some calculations

References

Index

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