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In complex analysis, Pólya's shire theorem, due to the mathematician George Pólya, describes the asymptotic distribution of the zeros of successive derivatives of a meromorphic function on the complex plane.

Statement of the theorem

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 location or order of each pole.