Contemporary Mathematics 2000; 367 pp; softcover Volume: 270 ISBN10: 0821826220 ISBN13: 9780821826225 List Price: US$105 Member Price: US$84 Order Code: CONM/270
 This book is the result of a meeting that took place at the University of Ghent (Belgium) on the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. Included are written articles detailing the lectures that were given as well as contributed papers on current topics of interest. The following areas are addressed: an historical overview of Hilbert's tenth problem, Hilbert's tenth problem for various rings and fields, model theory and localglobal principles, including relations between model theory and algebraic groups and analytic geometry, conjectures in arithmetic geometry and the structure of diophantine sets, for example with Mazur's conjecture, Lang's conjecture, and Bücchi's problem, and results on the complexity of diophantine geometry, highlighting the relation to the theory of computation. The volume allows the reader to learn and compare different approaches (arithmetical, geometrical, topological, modeltheoretical, and computational) to the general structural analysis of the set of solutions of polynomial equations. It would make a nice contribution to graduate and advanced graduate courses on logic, algebraic geometry, and number theory. Readership Graduate students, teachers, and research mathematicians working in logic, algebraic geometry, and number theory. Table of Contents  Y. Matiyasevich  Hilbert's tenth problem: What was done and what is to be done
 T. Pheidas and K. Zahidi  Undecidability of existential theories of rings and fields: A survey
 A. Shlapentokh  Hilbert's tenth problem over number fields, a survey
 M. Prunescu  Defining constant polynomials
 L. Darnière  Decidability and localglobal principles
 L. MoretBailly  Applications of localglobal principles to arithmetic and geometry
 J. Schmid  Regularly \(T\)closed fields
 M. Jarden, A. Razon, and W.D. Geyer  Skolem density problems over large Galois extensions of global fields
 T. Pheidas  An effort to prove that the existential theory of \(\mathbf Q\) is undecidable
 G. Cornelissen and K. Zahidi  Topology of Diophantine sets: Remarks on Mazur's conjectures
 P. Vojta  Diagonal quadratic forms and Hilbert's tenth problem
 J. M. Rojas  Algebraic geometry over four rings and the frontier to tractability
 A. Pillay  Some model theory of compact complex spaces
 K. H. Kim and F. W. Roush  Double coset decompositions for algebraic groups over \(K[t]\)
 C. D. Bennett, L. K. Elderbrock, and A. M. W. Glass  Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers
