The translation geometry of P\'olya's shires

Abstract

In his shire theorem, G. P\'olya proves that the zeros of iterated derivatives of a meromorphic function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. By recasting the local arguments of P\'olya into the language of translation surfaces, we prove its generalisation describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator $T_\omega: f \mapsto \frac{df}{\omega}$ where $\omega$ is a given meromorphic $1$-form. The accumulation set of these zeros is the union of edges of a generalised Voronoi diagram defined by the initial function $f$ together with the singular flat metric on the Riemann surface induced by $\omega$. This result provides the ground for a novel approach to the problem of finding a flat geometric presentation of a translation surface initially defined in terms of algebraic or complex-analytic data.