Distribution of critical points of polynomials

Abstract

The problem of the distribution of the critical points of polynomials in terms of the distribution μ \mu of the zeros is considered. It is shown that away from the inner boundary of the (compact) support S S of μ \mu the two distributions are the same. This is the case, in particular, if S S has connected complement. Examples are given showing that the two distributions may not be the same everywhere if the inner boundary has positive μ \mu -measure, but it is also shown that such examples are rare and very unstable.