Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry
About this Title
Jan Denef, Leonard Lipshitz, Thanases Pheidas and Jan Van Geel, Editors
Publication: Contemporary Mathematics
Publication Year : Volume 270
ISBNs: 978-0-8218-2622-5 (print); 978-0-8218-7860-6 (online)
MathSciNet review: 1802007
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.
Graduate students, teachers, and research mathematicians working in logic, algebraic geometry, and number theory.
Table of Contents
- Yuri Matiyasevich – Hilbert's tenth problem: what was done and what is to be done [MR 1802008]
- Thanases Pheidas and Karim Zahidi – Undecidability of existential theories of rings and fields: a survey [MR 1802009]
- Alexandra Shlapentokh – Hilbert's tenth problem over number fields, a survey [MR 1802010]
- Mihai Prunescu – Defining constant polynomials [MR 1802011]
- L. Darnière – Decidability and local-global principles [MR 1802012]
- Laurent Moret-Bailly – Applications of local-global principles to arithmetic and geometry [MR 1802013]
- Joachim Schmid – Regularly -closed fields [MR 1802014]
- Moshe Jarden and Aharon Razon – Skolem density problems over large Galois extensions of global fields [MR 1802015]
- Thanases Pheidas – An effort to prove that the existential theory of is undecidable [MR 1802016]
- Gunther Cornelissen and Karim Zahidi – Topology of Diophantine sets: remarks on Mazur's conjectures [MR 1802017]
- Paul Vojta – Diagonal quadratic forms and Hilbert's tenth problem [MR 1802018]
- J. Maurice Rojas – Algebraic geometry over four rings and the frontier to tractability [MR 1802019]
- Anand Pillay – Some model theory of compact complex spaces [MR 1802020]
- K. H. Kim and F. W. Roush – Double coset decompositions for algebraic groups over [MR 1802021]
- Curtis D. Bennett, Lisa K. Elderbrock and Andrew M. W. Glass – Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers [MR 1802022]