A restricted version of the Hilbert's 16th problem for quadratic vector fields

The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"\sigma$--distant from centers and $\ka$-distant from singular quadratic vector fields" provided that the limit cycles are $"\delta$--distant from singular points and infinity".