{"title":"A Geometric Study of Circle Packings and Ideal Class Groups","authors":"Daniel E. Martin","doi":"10.1007/s00454-024-00638-w","DOIUrl":null,"url":null,"abstract":"<p>A family of fractal arrangements of circles is introduced for each imaginary quadratic field <i>K</i>. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of <i>K</i> are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00638-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.
我们为每个虚二次域 K 引入了一个圆的分形排列族。总体而言，这些排列包含（直到仿射变换）扩展复平面中具有积分曲率和扎里斯基密集对称群的每个圆集。当这个集合是一个圆包装时，我们展示了我们的排列的环境结构是如何给出满足渐近局部-全局原理的几何标准的。我们还探讨了与 K 的类群的联系。其中有一个几何性质保证了某些理想类是群发电机。
期刊介绍：
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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