Restricted version of the infinitesimal Hilbert 16th problem

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.