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 Mosc. Math. J.: Year: Volume: Issue: Page: Find

 Mosc. Math. J., 2007, Volume 7, Number 2, Pages 281–325 (Mi mmj284)

Restricted version of the infinitesimal Hilbert 16th problem

A. A. Glutsyukab, Yu. S. Ilyashenkoc

a CNRS — Unit of Mathematics, Pure and Applied
b Laboratoire J.-V. Poncelet, Independent University of Moscow
c Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The paper deals with an abelian integral of a polynomial 1-form along a family of real ovals of a polynomial (hamiltonian) in two variables (the integral is considered as a function of value of the Hamiltonian). We give an explicit upper bound on the number of its zeroes (assuming the Hamiltonian ultra-Morse of arbitrary degree and ranging in a compact subset in the space of ultra-Morse polynomials of a given degree, and that the form has smaller degree). This bound depends on the choice of the compact set and is exponential in the fourth power of the degree.

Key words and phrases: Two-dimensional polynomial Hamiltonian vector field, oval, polynomial 1-form, Abelian integral, complex level curve, critical value, vanishing cycle.

DOI: https://doi.org/10.17323/1609-4514-2007-7-2-281-325

Full text: http://www.ams.org/.../abst7-2-2007.html
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Bibliographic databases:

MSC: Primary 58F21; 14K20; Secondary 34C05
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Citation: A. A. Glutsyuk, Yu. S. Ilyashenko, “Restricted version of the infinitesimal Hilbert 16th problem”, Mosc. Math. J., 7:2 (2007), 281–325

Citation in format AMSBIB
\Bibitem{GluIly07} \by A.~A.~Glutsyuk, Yu.~S.~Ilyashenko \paper Restricted version of the infinitesimal Hilbert 16th problem \jour Mosc. Math.~J. \yr 2007 \vol 7 \issue 2 \pages 281--325 \mathnet{http://mi.mathnet.ru/mmj284} \crossref{https://doi.org/10.17323/1609-4514-2007-7-2-281-325} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2337884} \zmath{https://zbmath.org/?q=an:1134.34019} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261829300009} 

• http://mi.mathnet.ru/eng/mmj284
• http://mi.mathnet.ru/eng/mmj/v7/i2/p281

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Ilyashenko Yu., “Some open problems in real and complex dynamical systems”, Nonlinearity, 21:7 (2008), T101–T107
2. Binyamini G., Yakovenko S., “Polynomial bounds for the oscillation of solutions of Fuchsian systems”, Ann Inst Fourier (Grenoble), 59:7 (2009), 2891–2926
3. Horozov E., Mihajlova A., “An improved estimate for the number of zeros of Abelian integrals for cubic Hamiltonians”, Nonlinearity, 23:12 (2010), 3053–3069
4. Binyamini G., Novikov D., Yakovenko S., “On the number of zeros of Abelian integrals”, Invent. Math., 181:2 (2010), 227–289