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Hardcover ISBN:  9780821828052 
Product Code:  CRMM/24 
List Price:  $115.00 
MAA Member Price:  $103.50 
AMS Member Price:  $92.00 
eBook ISBN:  9781470438685 
Product Code:  CRMM/24.E 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 
Hardcover ISBN:  9780821828052 
eBook ISBN:  9781470438685 
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 DetailsCRM Monograph SeriesVolume: 24; 2005; 192 ppMSC: 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 twentyfive 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 copublished with the Centre de recherches mathématiques.
ReadershipGraduate 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 HilbertArnol’d problem

Finiteness results in differential algebraic geometry and Diophantine geometry

Appendix A. ominimal structures, real analytic geometry, and transseries

Appendix B. List of lectures

Appendix C. Photographs of some workshop participants

Appendix D. List of participants


Additional Material

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This book focuses on finiteness conjectures and results in ordinary differential equations (ODEs) and Diophantine geometry. During the past twentyfive 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 copublished with the Centre de recherches mathématiques.
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 HilbertArnol’d problem

Finiteness results in differential algebraic geometry and Diophantine geometry

Appendix A. ominimal structures, real analytic geometry, and transseries

Appendix B. List of lectures

Appendix C. Photographs of some workshop participants

Appendix D. List of participants