A refinement of P\'olya's method to construct Voronoi diagrams for rational functions

Abstract

Given a complex polynomial $P$ with zeroes $z_1,\dotsc,z_d$, we show that the asymptotic zero-counting measure of the iterated derivatives $Q^{(n)}, \ n=1,2,\dotsc$, where $Q=R/P$ is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with $z_1,\dotsc,z_d$. This refines P\'olya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in $\mathbb C^m$.