  We consider the average number of limit cycles that bifurcate from a randomly
perturbed linear center where the perturbation consists of random (bivariate)
polynomials with independent coefficients. This problem reduces, by way of
classical perturbation theory of the Poincar\'e first return map, to a problem
on the real zeros of a random \emph{univariate} polynomial $\displaystyle
f_n(x) = \sum_{m=0}^n c_m \xi_m x^m$ with independent coefficients $\xi_m$
having mean zero, variance 1 and $c_m \sim m^{-1/2}$. This polynomial belongs
to the class of {\it generalized Kac polynomials} at the critical regime. We
provide asymptotics for the average number of real zeros and answer the
question on bifurcating limit cycles. Additionally, we provide the correct
order of the mean number of real roots in the subcritical regime.
