A refinement of P\'olya's method to construct Voronoi diagrams for rational functions
Rikard B\"ogvad, Christian H\"agg

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 with zeroes , we show that the asymptotic zero-counting measure of the iterated derivatives , where is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with . 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 .
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Taxonomy
TopicsMathematics and Applications · Mathematical functions and polynomials · Matrix Theory and Algorithms
