Hilbert's tenth problem : relations with arithmetic and algebraic geometry : Workshop on Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry, November 2-5, 1999, Ghent University, Belgium / Jan Denef [and three others], editors.

Format:

Book

Conference/Event

Contributor:

Denef, Jan, 1951- editor.

Conference Name:

Workshop on Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry (1999 : Ghent University, Belgium), issuing body.

Series:

Contemporary mathematics (American Mathematical Society) ; 270.

Contemporary mathematics, 0271-4132 ; 270

Language:

English

Subjects (All):

Hilbert's tenth problem.

Arithmetical algebraic geometry.

Geometry, Algebraic.

Physical Description:

1 online resource (384 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

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

Contents:

""Contents""; ""Introduction""; ""List of Participants""; ""Hilbert's tenth problem: What was done and what is to be done""; ""Undecidability of existential theories of rings and fields: A survey""; ""1. Introduction""; ""2. The existential theory of rational function fields""; ""3. The existential theory of algebraic function fields""; ""4. A geometric analogue of Hilbert's Tenth Problem""; ""5. Decision problems concerning rings of analytic functions: introduction""; ""6. First-order theories of rings of analytic functions""

""7. Existential decidability and the Approximation Property for rings of analytic functions""""8. Existential undecidability for rings of analytic functions""; ""9. Polynomial rings and p-adic entire functions""; ""10. Existential theories of fields of meromorphic functions""; ""11. Variations: Languages with predicates for symmetric functions""; ""Appendix A. Elliptic curves: Basic Definitions""; ""Appendix B. Elliptic curves over function fields""; ""Appendix C. Pell equations""; ""References""; ""Hilbert's tenth problem over number fields, a survey""; ""Defining constant polynomials""

""Decidability and local-global principles""""Applications of local-global principles to arithmetic and geometry""; ""Regularly T-closed fields""; ""Skolem density problems over large Galois extensions of global fields""; ""An effort to prove that the existential theory of Q is undecidable""; ""Topology of diophantine sets: Remarks on Mazur's conjectures""; ""Diagonal quadratic forms and Hilbert's tenth problem""; ""Algebraic geometry over four rings and the frontier to tractability""; ""1. Introduction""; ""2. Computing Complex Dimension Faster""

""3. Polytope Volumes and Counting Pieces of Semi-Algebraic Sets""""4. The Generalized Riemann Hypothesis and Detecting Rational Points""; ""5. Effective Siegel Versus Detecting Integral Points on Surfaces""; ""6. Proofs of Our Main Technical Results""; ""7. Acknowledgements""; ""Appendix: How the Examples Were Computed""; ""References""; ""Some model theory of compact complex spaces""; ""Double coset decompositions for algebraic groups over K[t]""; ""Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

0-8218-7860-3

0-8218-5606-5