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 be a meromorphic function on the complex plane with as its set of poles. If is the set of all zeros of all the successive derivatives , then the derived set (or the set of all limit points) is as follows:
- if has only one pole, then is empty.
- if , then coincides with the edges of the Voronoi diagram determined by the set of poles . In this case, if , the interior of each Voronoi cell consisting of the points closest to than any other point in is called the -shire.
The derived set is independent of the order of each pole.
References
- 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, 85âÂÂ92 (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