  Classes of polynomial differential equations of degree n are considered. An
explicit upper bound on the size of the coefficients are given which implies
that each equation in the class has exactly n complex periodic solutions. In
most of the classes the upper bound can be improved when we consider real
periodic solutions. We present a proof to a recent conjecture on the number of
periodic solutions. The results are used to give upper bounds for the number of
limit cycles of polynomial two-dimensional systems.
