Restricted version of the infinitesimal Hilbert 16th problem
Résumé
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.