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Centennial History of Hilbert’s 16th Problem
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by Yu. Ilyashenko PDF
Bull. Amer. Math. Soc. 39 (2002), 301-354 Request permission

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
  • MR Author ID: 190226
  • Email: yulij@math.cornell.edu, yulijs@mccme.ru
  • Received by editor(s): December 20, 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 39 (2002), 301-354
  • MSC (2000): Primary 34Cxx, 34Mxx, 37F75
  • DOI: https://doi.org/10.1090/S0273-0979-02-00946-1
  • MathSciNet review: 1898209