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.^[1] It has applications in Nevanlinna theory.^{[2]: 55, 62}

Statement

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

1. if ${\displaystyle f}$ has only one pole, then ${\displaystyle E'}$ is empty.
2. if ${\displaystyle |P|\geq 2}$, then ${\displaystyle E'}$ coincides with the edges of the Voronoi diagram determined by the set of poles ${\displaystyle P}$. In this case, if ${\displaystyle a\in P}$, the interior of each Voronoi cell consisting of the points closest to ${\displaystyle a}$ than any other point in ${\displaystyle P}$ is called the ${\displaystyle a}$-shire.^[3]

The derived set is independent of the order of each pole.^[3]: 32