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 {\displaystyle f} be a meromorphic function on the complex plane with P ≠ ∅ {\displaystyle P\neq \emptyset } as its set of poles. If E {\displaystyle E} is the set of all zeros of all the successive derivatives f ′ , f ″ , f ( 3 ) , … {\displaystyle f',f'',f^{(3)},\ldots } , then the derived set E ′ {\displaystyle E'} (or the set of all limit points) is as follows:
if f {\displaystyle f} has only one pole, then E ′ {\displaystyle E'} is empty. if | P | ≥ 2 {\displaystyle |P|\geq 2} , then E ′ {\displaystyle E'} coincides with the edges of the Voronoi diagram determined by the set of poles P {\displaystyle P} . In this case, if a ∈ P {\displaystyle a\in P} , the interior of each Voronoi cell consisting of the points closest to a {\displaystyle a} than any other point in P {\displaystyle P} is called the a {\displaystyle a} -shire. The derived set is independent of the order of each pole. In the special case of a rational function f {\displaystyle f} with at least two distinct poles, the theorem has a measure-theoretic refinement: the normalized zero-counting measures of f ( n ) {\displaystyle f^{(n)}} converge to an explicitly described probability measure supported on the edges of the Voronoi diagram of the poles.
References
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
