**CRM Monograph Series**

Volume: 24;
2005;
192 pp;
Hardcover

MSC: Primary 34; 14;
Secondary 11; 12; 32; 37; 58

Print ISBN: 978-0-8218-2805-2

Product Code: CRMM/24

List Price: $69.00

Individual Member Price: $55.20

**Electronic ISBN: 978-1-4704-3868-5
Product Code: CRMM/24.E**

List Price: $69.00

Individual Member Price: $55.20

#### Supplemental Materials

# On Finiteness in Differential Equations and Diophantine Geometry

Share this page *Editors and Authors: *
*Dana Schlomiuk; Dana Schlomiuk; Andreĭ A. Bolibrukh; Sergei Yakovenko; Vadim Kaloshin; Alexandru Buium*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book focuses on finiteness conjectures and results in ordinary
differential equations (ODEs) and Diophantine geometry. During the past
twenty-five years, much progress has been achieved on finiteness conjectures,
which are the offspring of the second part of Hilbert's 16th problem. Even in
its simplest case, this is one of the very few problems on Hilbert's list which
remains unsolved. These results are about existence and estimation of finite
bounds for the number of limit cycles occurring in certain families of
ODEs. The book describes this progress, the methods used (bifurcation theory,
asymptotic expansions, methods of differential algebra, or geometry) and the
specific results obtained. The finiteness conjectures on limit cycles are part
of a larger picture that also includes finiteness problems in other areas of
mathematics, in particular those in Diophantine geometry where remarkable
results were proved during the same period of time. There is a chapter devoted
to finiteness results in Diophantine geometry obtained by using methods of
differential algebra, which is a connecting element between these parallel
developments in the book.

The volume can be used as an independent study text for advanced undergraduates
and graduate students studying ODEs or applications of differential algebra to
differential equations and Diophantine geometry. It is also a good entry
point for researchers interested these areas, in particular, in limit cycles of
ODEs, and in finiteness problems.

Contributors to the volume include Andre&ibreve; A. Bolibrukh and Alexandru
Buium. Available from the AMS by A. Buium is Arithmetic Differential
Equations, as Volume 118 in the Mathematical Surveys and
Monographs series.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Table of Contents

# Table of Contents

## On Finiteness in Differential Equations and Diophantine Geometry

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Foreword ix10
- Finiteness problems in differential equations and Diophantine geometry 112
- Linear differential equations, Fuchsian inequalities and multiplicities of zeros 1122
- Quantitative theory of ordinary differential equations and the tangential Hilbert 16th problem 4152
- Around the Hilbert-Arnol’d problem 111122
- Finiteness results in differential algebraic geometry and Diophantine geometry 163174
- Appendix A. o-minimal structures, real analytic geometry, and transseries 175186
- Appendix B. List of lectures 177188
- Appendix C. Photographs of some workshop participants 179190
- Appendix D. List of participants 181192
- Back Cover Back Cover1194

#### Readership

Graduate students and research mathematicians interested in ordinary differential equations, differential algebra, and Diophantine geometry.