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 local-global 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 Bucchi'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, model-theoretical, 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.
Hilbert's tenth problem: What was done and what is to be done by Y. Matiyasevich Undecidability of existential theories of rings and fields: A survey by T. Pheidas and K. Zahidi Hilbert's tenth problem over number fields, a survey by A. Shlapentokh Defining constant polynomials by M. Prunescu Decidability and local-global principles by L. Darniere Applications of local-global principles to arithmetic and geometry by L. Moret-Bailly Regularly $T$-closed fields by J. Schmid Skolem density problems over large Galois extensions of global fields by M. Jarden, A. Razon, and W.-D. Geyer An effort to prove that the existential theory of $\mathbf Q$ is undecidable by T. Pheidas Topology of Diophantine sets: Remarks on Mazur's conjectures by G. Cornelissen and K. Zahidi Diagonal quadratic forms and Hilbert's tenth problem by P. Vojta Algebraic geometry over four rings and the frontier to tractability by J. M. Rojas Some model theory of compact complex spaces by A. Pillay Double coset decompositions for algebraic groups over $K[t]$ by K. H. Kim and F. W. Roush Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers by C. D. Bennett, L. K. Elderbrock, and A. M. W. Glass.