{"title":"对称半代数集合中的连通性","authors":"Cordian Riener, Robin Schabert, Thi Xuan Vu","doi":"arxiv-2404.09749","DOIUrl":null,"url":null,"abstract":"Semi-algebraic set is a subset of the real space defined by polynomial\nequations and inequalities. In this paper, we consider the problem of deciding\nwhether two given points in a semi-algebraic set are connected. We restrict to\nthe case when all equations and inequalities are invariant under the action of\nthe symmetric group and their degrees at most $d<n$, where $n$ is the number of\nvariables. Additionally, we assume that the two points are in the same\nfundamental domain of the action of the symmetric group, by assuming that the\ncoordinates of two given points are sorted in non-decreasing order. We\nconstruct and analyze an algorithm that solves this problem, by taking\nadvantage of the group action, and has a complexity being polynomial in $n$.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"38 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Connectivity in Symmetric Semi-Algebraic Sets\",\"authors\":\"Cordian Riener, Robin Schabert, Thi Xuan Vu\",\"doi\":\"arxiv-2404.09749\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Semi-algebraic set is a subset of the real space defined by polynomial\\nequations and inequalities. In this paper, we consider the problem of deciding\\nwhether two given points in a semi-algebraic set are connected. We restrict to\\nthe case when all equations and inequalities are invariant under the action of\\nthe symmetric group and their degrees at most $d<n$, where $n$ is the number of\\nvariables. Additionally, we assume that the two points are in the same\\nfundamental domain of the action of the symmetric group, by assuming that the\\ncoordinates of two given points are sorted in non-decreasing order. We\\nconstruct and analyze an algorithm that solves this problem, by taking\\nadvantage of the group action, and has a complexity being polynomial in $n$.\",\"PeriodicalId\":501033,\"journal\":{\"name\":\"arXiv - CS - Symbolic Computation\",\"volume\":\"38 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Symbolic Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2404.09749\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Symbolic Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2404.09749","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Semi-algebraic set is a subset of the real space defined by polynomial
equations and inequalities. In this paper, we consider the problem of deciding
whether two given points in a semi-algebraic set are connected. We restrict to
the case when all equations and inequalities are invariant under the action of
the symmetric group and their degrees at most $d
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