{"title":"交叉比度和三角测量","authors":"Rob Silversmith","doi":"10.1112/blms.13148","DOIUrl":null,"url":null,"abstract":"<p>The cross-ratio degree problem counts configurations of <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> points on <span></span><math>\n <semantics>\n <msup>\n <mi>P</mi>\n <mn>1</mn>\n </msup>\n <annotation>$\\mathbb {P}^1$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$n-3$</annotation>\n </semantics></math> prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-gon; these degrees are connected to the geometry of the real locus of <span></span><math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mrow>\n <mn>0</mn>\n <mo>,</mo>\n <mi>n</mi>\n </mrow>\n </msub>\n <annotation>$M_{0,n}$</annotation>\n </semantics></math>, and to positive geometry.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 11","pages":"3518-3529"},"PeriodicalIF":0.8000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13148","citationCount":"0","resultStr":"{\"title\":\"Cross-ratio degrees and triangulations\",\"authors\":\"Rob Silversmith\",\"doi\":\"10.1112/blms.13148\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The cross-ratio degree problem counts configurations of <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> points on <span></span><math>\\n <semantics>\\n <msup>\\n <mi>P</mi>\\n <mn>1</mn>\\n </msup>\\n <annotation>$\\\\mathbb {P}^1$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$n-3$</annotation>\\n </semantics></math> prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>-gon; these degrees are connected to the geometry of the real locus of <span></span><math>\\n <semantics>\\n <msub>\\n <mi>M</mi>\\n <mrow>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <annotation>$M_{0,n}$</annotation>\\n </semantics></math>, and to positive geometry.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 11\",\"pages\":\"3518-3529\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13148\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13148\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13148","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
交叉比度问题计算 P 1 $\mathbb {P}^1$ 上 n 个 $n$ 点的配置，其中有 n - 3 个 $n-3$ 规定的交叉比。交叉比度问题出现在组合学和几何的许多角落，但它们的结构一般还不太清楚。有趣的是，研究该问题的各种特例可以得到既多样又丰富的组合结构。在本文中，我们证明了一类以 n $n$ -gon 的三角形为索引的交叉比率度的简单封闭公式；这些度与 M 0 , n $M_{0,n}$ 的实部几何以及正几何相关联。
The cross-ratio degree problem counts configurations of points on with prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an -gon; these degrees are connected to the geometry of the real locus of , and to positive geometry.
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