{"title":"爪树的系统发生度","authors":"Rodica Andreea Dinu , Martin Vodička","doi":"10.1016/j.jcta.2024.105886","DOIUrl":null,"url":null,"abstract":"<div><p>Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called <em>phylogenetic degrees</em>. In this paper, we compute the phylogenetic degree of the variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with <span><math><mi>G</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span> and any <em>n</em>-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. We apply combinatorial methods and we give concrete formulas for these volumes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000256/pdfft?md5=ffe34f8c1bb09972f0075bfe4ea95627&pid=1-s2.0-S0097316524000256-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Phylogenetic degrees for claw trees\",\"authors\":\"Rodica Andreea Dinu , Martin Vodička\",\"doi\":\"10.1016/j.jcta.2024.105886\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called <em>phylogenetic degrees</em>. In this paper, we compute the phylogenetic degree of the variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with <span><math><mi>G</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span> and any <em>n</em>-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. We apply combinatorial methods and we give concrete formulas for these volumes.</p></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000256/pdfft?md5=ffe34f8c1bb09972f0075bfe4ea95627&pid=1-s2.0-S0097316524000256-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000256\",\"RegionNum\":2,\"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":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000256","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety with and any n-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes . We apply combinatorial methods and we give concrete formulas for these volumes.
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
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