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The translation geometry of Pólya's shires

Rikard Bøgvad, Boris Shapiro, Guillaume Tahar, Sangsan Warakkagun

TL;DR

The paper develops a unified framework linking translation-surface geometry with the asymptotic zero distribution of iterated differential operators on meromorphic functions. By introducing the operator $T_{\omega}: f \mapsto \frac{df}{\omega}$ on a compact Riemann surface $X$ endowed with a translation structure $\omega$, it proves that the limit set of zeros concentrates on the edges of a generalized Voronoi diagram determined by the principal polar locus and the flat metric. The asymptotic root-counting measure is described explicitly as $\frac{1}{A}\,\mu_{\omega,f} + \frac{1}{A} \sum_{p\in\mathcal{P}} (d_{p}-1)\delta_{p}$, where $A$ is the sum of $(a_z+1)$ over zeros of $\omega$ and $d_p$ are pole orders of $\omega$, embedding analytic data into flat-geometry structure. The work provides a novel pathway to flat geometric presentations of translation surfaces and connects complex-analytic data with explicit polygonal decompositions, offering broad applicability to derivatives of algebraic functions and potential extensions to more general connections and higher-order differentials.

Abstract

In his shire theorem, G. Pólya 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 Pólya 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 $T_ω: f \mapsto \frac{df}ω$ where $ω$ 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 $ω$. 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.

The translation geometry of Pólya's shires

TL;DR

The paper develops a unified framework linking translation-surface geometry with the asymptotic zero distribution of iterated differential operators on meromorphic functions. By introducing the operator on a compact Riemann surface endowed with a translation structure , it proves that the limit set of zeros concentrates on the edges of a generalized Voronoi diagram determined by the principal polar locus and the flat metric. The asymptotic root-counting measure is described explicitly as , where is the sum of over zeros of and are pole orders of , embedding analytic data into flat-geometry structure. The work provides a novel pathway to flat geometric presentations of translation surfaces and connects complex-analytic data with explicit polygonal decompositions, offering broad applicability to derivatives of algebraic functions and potential extensions to more general connections and higher-order differentials.

Abstract

In his shire theorem, G. Pólya 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 Pólya 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 where is a given meromorphic -form. The accumulation set of these zeros is the union of edges of a generalised Voronoi diagram defined by the initial function together with the singular flat metric on the Riemann surface induced by . 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.

Paper Structure

This paper contains 35 sections, 26 theorems, 114 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Short historical account
  3. Our set-up
  4. A first concrete example
  5. A second example: a monomial linear operator
  6. Two final examples in genus one.
  7. Organisation of the paper
  8. Primer on translation surfaces
  9. Translation structures
  10. Local models of zeros
  11. Geodesics and saddle connections
  12. Local models of simple poles
  13. Local models of poles of higher order
  14. Flat metric
  15. Translation surfaces are metrically complete
  16. ...and 20 more sections

Key Result

Theorem 1.5

Consider a non-zero meromorphic $1$-form $\omega$ on a compact Riemann surface $X$, its associated differential operator $T_{\omega}$ and any meromorphic function $f$ on $X$ such that $\mathcal{PPL}(\omega,f) \neq \emptyset$. Then the following two facts are valid. (i) The limit set $\mathcal{L}(T_{ (ii) The asymptotic root-counting measure of the sequence $(T_{\omega}^{n} (f))_{n \in \mathbb{N}}$

Figures (12)

  • Figure 1: The Voronoi diagram determined by the five poles (red triangles) of some rational function $f$, together with the zeros of $f^{(20)}$ in blue dots.
  • Figure 2: Zeros of $f^{(40)}(z)$ lie on the real line $\mathbb{R}$ in blue dots. The asymptotic Cauchy measure of the segment $\alpha$ is the proportion out of $2\pi$ of the angle $\theta$ from either pole $\pm i$ (in red squares) subtended by the segment.
  • Figure 3: The lemniscate with the roots of $T_4^{90}\left(-\frac{1}{z+1}\right)$ accumulating on one of its ovals.
  • Figure 4: Let $\Lambda = \mathbb{Z} + e^{\frac{i\pi}{3}}\mathbb{Z}$ and $\omega = dz$. Left: The Voronoi diagram determined by the poles of $\wp(z)$ (red dots) consists of the blue dashed lines. Right: The zeros of $\wp^{(45)}$ are in blue dots. In this case, parallelograms cut out by the green lines are fundamental domains of the action of $\Lambda$ on $\mathbb{C}$.
  • Figure 5: Let $\Lambda = \mathbb{Z} + e^{\frac{i \pi}{3}}\mathbb{Z}$. Left: The red dots are the points of $\mathcal{PPL}(\omega, f)$, where $\omega = \wp_z(z)dz$ and $f(z) = \wp_z(z)$, which determine the Voronoi diagram in blue dashed lines. The green lines cut out fundamental domains of the action of $\Lambda$ on $\mathbb{C}$. The white circles in the corners are the poles of $\omega$. Right: The dark blue dots are the zeros of $T_{\omega}^{25}(f)$ in $\mathbb{C}$.
  • ...and 7 more figures

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 63 more