# Restricted version of the infinitesimal Hilbert 16th problem

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 of the number of its zeros (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 a smaller degree). This bound depends on the choice of the compact set and is exponential in the fourth power of the degree.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00078858
Contributor : Alexey Glutsyuk <>
Submitted on : Wednesday, June 7, 2006 - 5:11:41 PM
Last modification on : Wednesday, September 4, 2019 - 5:36:02 PM
Long-term archiving on : Monday, April 5, 2010 - 10:28:11 PM

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• HAL Id : ensl-00078858, version 1

### Citation

Alexey Glutsyuk, Yuli Ilyashenko. Restricted version of the infinitesimal Hilbert 16th problem. 2006. ⟨ensl-00078858⟩

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