The translation geometry of Polya’s shires

In his shire theorem, Polya proves that the zeros of iterated derivatives of a rational function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. Recasting the local arguments of Polya into the language of translation surfaces, we prove a generalization describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator defined by meromorphic 1-form. The accumulation set of these zeros is the union of edges of a generalized Voronoi diagram defined jointly by the initial function and the singular flat metric on the Riemann surface induced by the differential. This process offers a completely novel approach to the practical problem of finding a flat geometric presentation (a polygon with identification of pairs of edges) of a translation surface defined in terms of algebraic or complex-analytic data. This is a joint work with Rikard Bogvad, Boris Shapiro and Sangsan Warakkagun.

See webpage for more details: https://people.math.harvard.edu/~ctm/sem/