An extension of Pólya's shire theorem

Abstract

The classical Pólya's Shire Theorem states that for a meromorphic function $f$ in $\mathbb{C}$ with the set $S$ of its poles, the zeros of its iterated derivatives $f^{(n)}$ asymptotically accumulate when $n\to \infty$ along the edges of the Voronoi diagram associated with $S$. We extend Pólya's Shire Theorem to the differential operator with Laurent monomial coefficients; $\mathcal{D}_{-\ell}:= z^{-\ell} \smash{\frac{\partial }{\partial z}} $, when $\ell=1,2,3,\dots$, acting iteratively on rational functions with a single pole $r(z):={1}/{(z-b)}$ and to the differential operator with monomial coefficients of degree $\ell\geq 3$; $\mathcal{D}_{\ell}:= z^{\ell} \frac{\partial }{\partial z},\ell =3,4,5,\dots$, acting iteratively on arbitrary rational functions with $k\geq 2$ distinct simple poles of the form $h(z):=\sum^{k}_{i=1} {\alpha_i}/{(z-z_i)}$, respectively.

The results show that in the first case the zero loci of iterations of $D_{-\ell}(1/(z-b))$ concentrate on one part of the hyperbola (which contains the pole $b$), and that in the second case the zero loci of iterations of $D_{\ell} (h(z))$ concentrate on a certain part of the collection of the lemniscates in the $z$-plane. Moreover, the map $w(z)=z^{1-\ell}/(1-\ell)$, $\ell \geq 3$ sends the limiting set of the zeros of $\mathcal{D}^{n}_{\ell}(h(z))$ homeomorphically onto the edges of the Voronoi diagram in the $w$-plane.