Polya'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 $f', f'', f^, \ldots$, then the derived set $E'$ (or the set of all limit points) is as follows:

# if $f$ has only one pole, then $E'$ is empty.
# 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.

References

{{reflist
*Rikard Bögvad, Christian Hägg, A refinement of Pólya's method to construct Voronoi diagrams for rational functions, https://arxiv.org/abs/1610.00921

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

Theorems in complex analysis
Meromorphic functions