Pólya's shire theorem

Pólya's shire theorem, named after George Pólya, is a theorem in complex analysis that describes the asymptotic distribution of the zeros of successive derivatives of a meromorphic function on the complex plane. It has applications in Nevanlinna theory.

Statement

Let $f$ be a meromorphic function on the complex plane with $P\neq \emptyset$ as its set of poles. If $E$ is the set of all zeros of all the successive derivatives $f',f'',f^{(3)},\ldots$, then the derived set $E'$ (or the set of all limit points) is as follows:

1. if $f$ has only one pole, then $E'$ is empty.
2. if $|P|\geq 2$, then $E'$ coincides with the edges of the Voronoi diagram determined by the set of poles $P$. In this case, if $a\in P$, the interior of each Voronoi cell consisting of the points closest to $a$ than any other point in $P$ is called the $a$-shire.

The derived set is independent of the order of each pole.

In the special case of a rational function $f$ with at least two distinct poles, the theorem has a measure-theoretic refinement: the normalized zero-counting measures of $f^{(n)}$ converge to an explicitly described probability measure supported on the edges of the Voronoi diagram of the poles.

Further reading

• Weiss, M. "Pólya's Shire Theorem for Automorphic Functions". Geometriae Dedicata 100, 85–92 (2003). https://doi.org/10.1023/A:1025855513977
• Robert M. Gethner, A Pólya "shire" Theorem for Entire Functions. University of Wisconsin-Madison, (1982) https://www.google.com/books/edition/A_P%C3%B3lya_shire_Theorem_for_Entire_Functi/NmBxAAAAMAAJ

• https://qzc.tsinghua.edu.cn/info/1192/5825.htm
• https://link.springer.com/article/10.1023/A:1025855513977