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On Finiteness in Differential Equations and Diophantine Geometry
 
Dana Schlomiuk Université de Montréal, Montréal, QC, Canada
Andreĭ A. Bolibrukh Steklov Institute, Russian Academy of Sciences, Moscow, Russia
Sergei Yakovenko Weizmann Institute of Science, Rehovot, Israel
Vadim Kaloshin California Institute of Technology, Pasadena, CA
Alexandru Buium University of New Mexico, Albuquerque, NM
Edited by: Dana Schlomiuk Université de Montréal, Montréal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques
On Finiteness in Differential Equations and Diophantine Geometry
Hardcover ISBN:  978-0-8218-2805-2
Product Code:  CRMM/24
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBook ISBN:  978-1-4704-3868-5
Product Code:  CRMM/24.E
List Price: $110.00
MAA Member Price: $99.00
AMS Member Price: $88.00
Hardcover ISBN:  978-0-8218-2805-2
eBook: ISBN:  978-1-4704-3868-5
Product Code:  CRMM/24.B
List Price: $225.00 $170.00
MAA Member Price: $202.50 $153.00
AMS Member Price: $180.00 $136.00
On Finiteness in Differential Equations and Diophantine Geometry
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On Finiteness in Differential Equations and Diophantine Geometry
Dana Schlomiuk Université de Montréal, Montréal, QC, Canada
Andreĭ A. Bolibrukh Steklov Institute, Russian Academy of Sciences, Moscow, Russia
Sergei Yakovenko Weizmann Institute of Science, Rehovot, Israel
Vadim Kaloshin California Institute of Technology, Pasadena, CA
Alexandru Buium University of New Mexico, Albuquerque, NM
Edited by: Dana Schlomiuk Université de Montréal, Montréal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques
Hardcover ISBN:  978-0-8218-2805-2
Product Code:  CRMM/24
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBook ISBN:  978-1-4704-3868-5
Product Code:  CRMM/24.E
List Price: $110.00
MAA Member Price: $99.00
AMS Member Price: $88.00
Hardcover ISBN:  978-0-8218-2805-2
eBook ISBN:  978-1-4704-3868-5
Product Code:  CRMM/24.B
List Price: $225.00 $170.00
MAA Member Price: $202.50 $153.00
AMS Member Price: $180.00 $136.00
  • Book Details
     
     
    CRM Monograph Series
    Volume: 242005; 192 pp
    MSC: Primary 34; 14; Secondary 11; 12; 32;

    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 Andreĭ 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.

    Readership

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

  • Table of Contents
     
     
    • Chapters
    • Finiteness problems in differential equations and Diophantine geometry
    • Linear differential equations, Fuchsian inequalities and multiplicities of zeros
    • Quantitative theory of ordinary differential equations and the tangential Hilbert 16th problem
    • Around the Hilbert-Arnol’d problem
    • Finiteness results in differential algebraic geometry and Diophantine geometry
    • Appendix A. o-minimal structures, real analytic geometry, and transseries
    • Appendix B. List of lectures
    • Appendix C. Photographs of some workshop participants
    • Appendix D. List of participants
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 242005; 192 pp
MSC: Primary 34; 14; Secondary 11; 12; 32;

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 Andreĭ 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.

Readership

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

  • Chapters
  • Finiteness problems in differential equations and Diophantine geometry
  • Linear differential equations, Fuchsian inequalities and multiplicities of zeros
  • Quantitative theory of ordinary differential equations and the tangential Hilbert 16th problem
  • Around the Hilbert-Arnol’d problem
  • Finiteness results in differential algebraic geometry and Diophantine geometry
  • Appendix A. o-minimal structures, real analytic geometry, and transseries
  • Appendix B. List of lectures
  • Appendix C. Photographs of some workshop participants
  • Appendix D. List of participants
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.