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| Acknowledgements | |
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| Introduction | |
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| In the beginning | |
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| Diophantine definitions and Diophantine sets | |
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| Diophantine classes: definitions and basic facts | |
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| Diophantine generation | |
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| Diophantine generation of integral closure and Dioph-regularity | |
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| Big picture: Diophantine family of a ring | |
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| Diophantine equivalence and Diophantine decidability | |
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| Weak presentations | |
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| Some properties of weak presentations | |
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| How many Diophantine classes are there? | |
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| Diophantine generation and Hilbert's Tenth Problem | |
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| Integrality at finitely many primes and divisibility of order at infinitely many primes | |
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| The main ideas | |
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| Integrality at finitely many primes in number fields | |
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| Integrality at finitely many primes over function fields | |
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| Divisibility of order at infinitely many primes over number fields | |
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| Divisibility of order at infinitely many primes over function fields | |
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| Bound equations for number fields and their consequences | |
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| Real embeddings | |
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| Using divisibility in the rings of algebraic integers | |
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| Using divisibility in bigger rings | |
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| Units of rings of W-integers of norm 1 | |
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| What are the units of the rings of W-integers? | |
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| Norm equations of units | |
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| The Pell equation | |
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| Non-integral solutions of some unit norm equations | |
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| Diophantine classes over number fields | |
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| Vertical methods of Denef and Lipshitz | |
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| Integers of totally real number fields and fields with exactly one pair of non-real embeddings | |
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| Integers of extensions of degree 2 of totally real number fields | |
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| The main results for the rings of W-integers and an overview of the proof | |
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| The main vertical definability results for rings of W-integers in totally real number fields | |
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| Consequences for vertical definability over totally real fields | |
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| Horizontal definability for rings of W-integers of totally real number fields and Diophantine undecidability for these rings | |
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| Vertical definability results for rings of W-integers of the totally complex extensions of degree 2 of totally real number fields | |
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| Some consequences | |
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| Big picture for number fields revisited | |
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| Further results | |
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| Diophantine undecidability of function fields | |
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| Defining multiplication through localized divisibility | |
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| pth power equations over function fields I: Overview and preliminary results | |
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| pth power equations over function fields II: pth powers of a special element | |
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| pth power equations over function fields III: pth powers of arbitrary functions | |
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| Diophantine model of Z over function fields over finite fields of constants | |
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| Bounds for function fields | |
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| Height bounds | |
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| Using pth powers to bound the height | |
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| Diophantine classes over function fields | |
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| The weak vertical method revisited | |
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| Weak vertical method applied to non-constant cyclic extensions | |
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| The weak vertical method applied to constant field extensions | |
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| Vertical definability for large subrings of global function fields | |
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| Integrality at infinitely many primes over global function fields | |
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| The big picture for function fields revisited | |
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| Mazur's conjectures and their consequences | |
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| The two conjectures | |
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| A ring version of Mazur's first conjecture | |
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| First counterexamples | |
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| Consequences for Diophantine models | |
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| Results of Poonen | |
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| A statement of the main theorem and an overview of the proof | |
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| Properties of elliptic curves I: Factors of denominators of points | |
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| Properties of elliptic curves II: The density of set of "largest" primes | |
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| Properties of elliptic curves III: Finite sets looking big | |
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| Properties of elliptic curves IV: Consequences of a result of Vinograd | |
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| Construction of sets T[subscript 1](P) and T[subscript 2](P) and their properties | |
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| Proof of Poonen's theorem | |
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| Beyond global fields | |
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| Function fields of positive characteristic and of higher transcendence degree or over infinite fields of constants | |
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| Algebraic extensions of global fields of infinite degree | |
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| Function fields of characteristic 0 | |
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| Recursion (computability) theory | |
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| Computable (recursive) functions | |
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| Recursively enumerable sets | |
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| Turing and partial degrees | |
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| Degrees of sets of indices, primes, and products | |
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| Recursive algebra | |
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| Recursive presentation of Q | |
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| Recursive presentation of other fields | |
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| Representing sets of primes and rings of S-integers in number fields | |
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| Number theory | |
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| Global fields, valuations, and rings of W-integers | |
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| Existence through approximation theorems | |
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| Linearly disjoint fields | |
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| Divisors, prime and composite, under extensions | |
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| Density of prime sets | |
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| Elliptic curves | |
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| Coordinate polynomials | |
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| Basic facts about local fields | |
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| Derivations | |
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| Some calculations | |
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| References | |
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| Index | |