A refinement for rational functions of Pólya's method to construct Voronoi diagrams

Given a complex polynomial P with zeroes z1,…,zd, we show that the asymptotic zero-counting measure of the iterated derivatives Q(n), n=1,2,…, where Q=R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z1,…,zd. This refines Pólya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in Cm.