Abstract：

Pólya's classical Shire Theorem states that the zeros of the successive derivatives of a meromorphic function on the complex plane accumulate onto the edges of the Voronoi diagram determined by the loci of the poles of the function. We develop a generalization to describe the limit set of the zeros of the iterates of a meromorphic function on a compact Riemann surface under a linear differential operator defined by a meromorphic 1-form. Refining Pólya's local arguments, we show that the accumulation set is the union of the edges of a generalized Voronoi diagram defined by the meromorphic function and the singular flat metric induced by the 1-form. This is ongoing work in progress with Boris Shapiro and Guillaume Tahar.