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

Abstract

Given a complex polynomial $P$ with zeroes $z_1,\dots ,z_d$, we show that the asymptotic zero-counting measure of the iterated derivatives $Q^{(n)},n=1,2,\dots$, where $Q=R/P$ is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with $z_1,\dots ,z_d$. 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 ${\mathbb{C}}^m$.

Final version as accepted by JMAA, with some minor corrections

References in corpus (2)

• Complex Tropical Currents, Extremality, and Approximations
• On mother body measures with algebraic Cauchy transform