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

TL;DR

This paper refines Pólya's method to show that the zero distribution of derivatives of certain rational functions converges to a measure supported on Voronoi diagrams, extending to hyperplane configurations in complex space.

Contribution

It provides a refined approach to Pólya's theorem, explicitly constructing measures supported on Voronoi diagrams for rational functions and hyperplane arrangements.

Findings

Zero-counting measures of derivatives converge to Voronoi-supported measures.

Extension of results to hyperplane configurations in complex spaces.

Explicit construction of probability measures on Voronoi diagrams.

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 $\mathbb C^m$.