This article is cited in 4 scientific papers (total in 4 papers)
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
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.
MSC: Primary 58F21; 14K20; Secondary 34C05
Received: May 24, 2006
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
\by A.~A.~Glutsyuk, Yu.~S.~Ilyashenko
\paper Restricted version of the infinitesimal Hilbert 16th problem
\jour Mosc. Math.~J.
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Ilyashenko Yu., “Some open problems in real and complex dynamical systems”, Nonlinearity, 21:7 (2008), T101–T107
Binyamini G., Yakovenko S., “Polynomial bounds for the oscillation of solutions of Fuchsian systems”, Ann Inst Fourier (Grenoble), 59:7 (2009), 2891–2926
Horozov E., Mihajlova A., “An improved estimate for the number of zeros of Abelian integrals for cubic Hamiltonians”, Nonlinearity, 23:12 (2010), 3053–3069
Binyamini G., Novikov D., Yakovenko S., “On the number of zeros of Abelian integrals”, Invent. Math., 181:2 (2010), 227–289
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