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

An account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields.

Alexandra Shlapentokh (Author)

9780521833608, Cambridge University Press

Hardback, published 9 November 2006

330 pages, 18 b/w illus.
23.3 x 15.8 x 2.3 cm, 0.588 kg

"It gives a very comprehensive survey of what is known so far about undecidability and Diophantine definability for these rings..."
Jeroen Demeyer, Mathematical Reviews

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 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.

1. Introduction
2. Diophantine classes: definition and basic facts
3. Diophantine equivalence and diophantine decidability
4. Integrality at finitely many primes and divisibility of order at infinitely many primes
5. Bound equations for number fields and their consequences
6. Units of rings of W-integers of norm 1
7. Diophantine classes over number fields
8. Diophantine undecidability of function fields
9. Bounds for function fields
10. Diophantine classes over function fields
11. Mazur's conjectures and their consequences
12. Results of Poonen
13. Beyond global fields
A. Recursion theory
B. Number theory
Bibliography
Index.

Subject Areas: Number theory [PBH]