Abstract
We estimate the expected number of limits cycles of a planar polynomial vetor field situated in a neighbourhood of the origin provided that the field in a large neighbourhood is close enough to a linear center. Our main tool is a distribution inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of pluripotential theory. This inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros.
