Abstract:

In his shire theorem, Polya proves that the zeros of iterated derivatives of a rational function in thecomplex 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 ageneralization describing the asymptotic distribution of the zerosof a meromorphic function on acompact Riemann surface under the iterations of a linear differential operator defined by meromor-phic 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 in-duced by the differential. This process offers a completely novel approach to the practical problemof 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 BorisShapiro and Sangsan Warakkagun.