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 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, 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.

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 local-global principles
• L. Moret-Bailly -- Applications of local-global 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