Hilbert's Tenth Problem

Alexandra Shlapentokh is Professor of Mathematics at East Carolina University.

• 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.

"Shlapentokh offers the first synthesis of the new wave of work, including exciting recent results of B.Poonen that come as close as anyone can yet to extending Matiyesevich's theorem in the desired way." D.V. Feldman, University of New Hampshire for CHOICE