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

Cambridge University Press, 2006 — 330 p. — (New Mathematical Monographs: 7). — ISBN: 978-0-521-83360-8.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.Looks at the subject from the point of view of Algebraic Number Theory.
Also includes information on Mazur's Conjectures and Poonen's elliptic curve method.
Suitable for graduate students.Diophantine classes: definition and basic facts
Diophantine equivalence and diophantine decidability
Integrality at finitely many primes and divisibility of order at infinitely many primes
Bound equations for number fields and their consequences
Units of rings of W-integers of norm 1
Diophantine classes over number fields
Diophantine undecidability of function fields
Bounds for function fields
Diophantine classes over function fields
Mazur's conjectures and their consequences
Results of Poonen
Beyond global fields
Recursion theory
Number theory