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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 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.

Hilbert's tenth problem. --- Mathematics. --- Math --- Science --- Decision problem of Diophantine sets --- Diophantine equations, Solvability problem of --- Diophantine sets, Decision problem of --- Solvability problem of Diophantine equations --- Diophantine analysis --- Algebraic number theory. --- Diophantine equations. --- Diophantic equations --- Equations, Diophantic --- Equations, Diophantine --- Equations, Indefinite --- Equations, Indeterminate --- Indefinite equations --- Indeterminate equations --- Number theory

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Hilbert's tenth problem. --- Computable functions. --- Computability theory --- Functions, Computable --- Partial recursive functions --- Recursive functions, Partial --- Constructive mathematics --- Decidability (Mathematical logic) --- Decision problem of Diophantine sets --- Diophantine equations, Solvability problem of --- Diophantine sets, Decision problem of --- Solvability problem of Diophantine equations --- Diophantine analysis