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Bulletin of the American Mathematical Society

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Centennial History of Hilbert's 16th Problem

Author: Yu. Ilyashenko
Journal: Bull. Amer. Math. Soc. 39 (2002), 301-354
MSC (2000): Primary 34Cxx, 34Mxx, 37F75
Published electronically: April 9, 2002
MathSciNet review: 1898209
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Abstract: The second part of Hilbert's 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below.

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Additional Information

Yu. Ilyashenko
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Address at time of publication: Moscow State and Independent Universities, Steklov Mathematical Institute, Moscow (MIAN) Gubkina st. 8, Moscow, Russia, 117966

Keywords: Limit cycles, polynomial vector fields, normal forms, bifurcations, foliations, Abelian integrals
Received by editor(s): December 1, 2001
Published electronically: April 9, 2002
Additional Notes: The author was supported in part by grants NSF DMS 997-0372, NSF 0010404, and CRDF RM1-2086. The main results of the paper were presented at colloquium talks at Cornell University, December 1999, and Northeastern University (Harvard - MIT - Brandeis - Northeastern Colloquium), November 2000. The author thanks Dr. S. Gelfand, who assisted with the latter talk and suggested the idea of writing a survey on the subject
Article copyright: © Copyright 2002 American Mathematical Society