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Hilbert's Tenth Problem Diophantine Classes and Extensions to Global Fields

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ISBN-10: 0521833604

ISBN-13: 9780521833608

Edition: 2007

Authors: Alexandra Shlapentokh, Bela Bollobas, William Fulton, Anatole Katok, Frances Kirwan

List price: $156.00
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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…    
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Book details

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

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
Some calculations