Polya's shire theorem

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

1. if $f$ has only one pole, then $E\text{'}$ is empty.
2. if $|P|\; \backslash geq\; 2$, then $E\text{'}$ coincides with the edges of the Voronoi diagram determined by the set of poles $P$. In this case, if $a\; \backslash in\; P$, the interior of each Voronoi cell consisting of the points closest to $a$ than any other point in $P$ is called the $a$-shire.{{Cite book |last=Whittaker |first=J.M. |author-link=John Macnaghten Whittaker |title=Interpolatory Function Theory |date=1935 |publisher=Cambridge University Press |pages=32–38 |url=https://archive.org/details/in.ernet.dli.2015.77626/page/n41/mode/2up}}

The derived set is independent of the order of each pole.{{rp|p=32}}

{{reflist}}

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

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

Category:Theorems in complex analysis

Category:Meromorphic functions