  In this note we initiate the probabilistic study of the critical points of
polynomials of large degree with a given distribution of roots. Namely, let f
be a polynomial of degree n whose zeros are chosen IID from a probability
measure mu on the complex numbers. We conjecture that the zero set of f' always
converges in distribution to mu as n goes to infinity. We prove this for
measures with finite one-dimensional energy. When mu is uniform on the unit
circle this condition fails. In this special case the zero set of f' converges
in distribution to that the IID Gaussian random power series, a well known
determinantal point process.
