**Contemporary Mathematics**

Volume: 270;
2000;
367 pp;
Softcover

MSC: Primary 00; 03; 11; 14; 65; 68;

Print ISBN: 978-0-8218-2622-5

Product Code: CONM/270

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

**Electronic ISBN: 978-0-8218-7860-6
Product Code: CONM/270.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

# Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry

Share this page *Edited by *
*Jan Denef; Leonard Lipshitz; Thanases Pheidas; Jan Van Geel*

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

## Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry

- Contents vii8 free
- Introduction ix10 free
- List of Participants xi12 free
- Hilbert's tenth problem: What was done and what is to be done 114 free
- Undecidability of existential theories of rings and fields: A survey 4962
- 1. Introduction 4962
- 2. The existential theory of rational function fields 6073
- 3. The existential theory of algebraic function fields 7184
- 4. A geometric analogue of Hilbert's Tenth Problem 7487
- 5. Decision problems concerning rings of analytic functions: introduction 7790
- 6. First-order theories of rings of analytic functions 7891
- 7. Existential decidability and the Approximation Property for rings of analytic functions 7992
- 8. Existential undecidability for rings of analytic functions 8194
- 9. Polynomial rings and p-adic entire functions 8598
- 10. Existential theories of fields of meromorphic functions 8699
- 11. Variations: Languages with predicates for symmetric functions 87100
- Appendix A. Elliptic curves: Basic Definitions 89102
- Appendix B. Elliptic curves over function fields 90103
- Appendix C. Pell equations 93106
- References 98111

- Hilbert's tenth problem over number fields, a survey 107120
- Defining constant polynomials 139152
- Decidability and local-global principles 145158
- Applications of local-global principles to arithmetic and geometry 169182
- Regularly T-closed fields 187200
- Skolem density problems over large Galois extensions of global fields 213226
- An effort to prove that the existential theory of Q is undecidable 237250
- Topology of diophantine sets: Remarks on Mazur's conjectures 253266
- Diagonal quadratic forms and Hilbert's tenth problem 261274
- Algebraic geometry over four rings and the frontier to tractability 275288
- 1. Introduction 276289
- 2. Computing Complex Dimension Faster 280293
- 3. Polytope Volumes and Counting Pieces of Semi-Algebraic Sets 287300
- 4. The Generalized Riemann Hypothesis and Detecting Rational Points 290303
- 5. Effective Siegel Versus Detecting Integral Points on Surfaces 294307
- 6. Proofs of Our Main Technical Results 299312
- 7. Acknowledgements 313326
- Appendix: How the Examples Were Computed 313326
- References 316329

- Some model theory of compact complex spaces 323336
- Double coset decompositions for algebraic groups over K[t] 339352
- Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers 357370