Multiple Extremal Disc-Packings in Compact Hyperbolic Surfaces

IF 0.7 4区数学 Q2 MATHEMATICS Experimental Mathematics Pub Date : 2022-06-08 DOI:10.1080/10586458.2022.2075491

Ernesto Girondo, Cristian Reyes

引用次数: 0

Abstract

The radius of a packing of metric discs embedded in a compact hyperbolic surface is bounded by an extremal value dependent upon the topology of the surface and the number of discs in the packing. In this paper we discuss the possibility of finding multiple extremal disc-packings within a given surface, determining the combinatorial-arithmetic conditions on the topology of the surface and the number of discs of the packing that allow such a phenomenon to happen. Moreover, we provide explicit examples of surfaces containing multiple extremal packings for each type of packing and each topological type of surface possible. Our construction relies in computer experimentation in two ways: first, by performing numerical computations that suggest certain surfaces as good candidates to contain more than one extremal packing, and second by checking with computer algebra software some lengthy necessary algebraic conditions in certain number fields that prove that the surfaces numerically constructed do indeed contain multiple extremal disc-packings.

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紧双曲曲面上的多极值圆盘填料

摘要:紧致双曲曲面上的公制圆盘的半径由一个极值限定，该极值取决于曲面的拓扑结构和圆盘的数量。本文讨论了在给定曲面上找到多个极值盘形填料的可能性，确定了曲面拓扑结构上的组合算法条件和填料的盘数，使这种现象得以发生。此外，我们为每种类型的填料和每种可能的拓扑类型的表面提供了包含多个极端填料的明确示例。我们的构造以两种方式依赖于计算机实验:第一，通过执行数值计算，建议某些表面作为包含多个极值填料的良好候选者，第二，通过计算机代数软件检查某些数域中一些冗长的必要代数条件，证明数值构造的表面确实包含多个极值圆盘填料。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.

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