Bosco Nyandwi: An Extension of Polya'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 \(S\). In this talk, we extend Pólya’s shire theorem to the differential operator with Laurent monomial coefficients; \(\mathcal{D}_{-\ell} := z^{-\ell} \frac{\partial}{\partial z}\), when \(\ell = 1,2,3,\dots\), acting iteratively on rational functions with a single pole \(r(z) := \frac{−1}{z+b}\) 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 \(k\) distinct simple poles of the form \(h(z) := \sum_{i=1}^k \frac{\alpha_i}{z−z_i}\) respectively. The results show that in both cases the zero loci of iterations concentrate on a certain part of collection of the lemniscates.