{"title":"对称分解和Veronese构造","authors":"Katharina Jochemko","doi":"10.1093/IMRN/RNAB031","DOIUrl":null,"url":null,"abstract":"We study rational generating functions of sequences $\\{a_n\\}_{n\\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\\{a_{rn}\\}_{n\\geq 0}$. We prove that if the numerator polynomial for $\\{a_n\\}_{n\\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for $\\{a_{rn}\\}_{n\\geq 0}$ is real-rooted whenever $r\\geq \\max \\{s,d+1-s\\}$. Moreover, if the numerator polynomial for $\\{a_n\\}_{n\\geq 0}$ is symmetric then we show that the symmetric decomposition for $\\{a_{rn}\\}_{n\\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\\ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\\geq \\max \\{s,d+1-s\\}$. If the polytope is Gorenstein then this decomposition is moreover interlacing.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Symmetric Decompositions and the Veronese Construction\",\"authors\":\"Katharina Jochemko\",\"doi\":\"10.1093/IMRN/RNAB031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study rational generating functions of sequences $\\\\{a_n\\\\}_{n\\\\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\\\\{a_{rn}\\\\}_{n\\\\geq 0}$. We prove that if the numerator polynomial for $\\\\{a_n\\\\}_{n\\\\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for $\\\\{a_{rn}\\\\}_{n\\\\geq 0}$ is real-rooted whenever $r\\\\geq \\\\max \\\\{s,d+1-s\\\\}$. Moreover, if the numerator polynomial for $\\\\{a_n\\\\}_{n\\\\geq 0}$ is symmetric then we show that the symmetric decomposition for $\\\\{a_{rn}\\\\}_{n\\\\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\\\\ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\\\\geq \\\\max \\\\{s,d+1-s\\\\}$. If the polytope is Gorenstein then this decomposition is moreover interlacing.\",\"PeriodicalId\":8442,\"journal\":{\"name\":\"arXiv: Combinatorics\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-04-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1093/IMRN/RNAB031\",\"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: Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1093/IMRN/RNAB031","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Symmetric Decompositions and the Veronese Construction
We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\{a_{rn}\}_{n\geq 0}$. We prove that if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for $\{a_{rn}\}_{n\geq 0}$ is real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is symmetric then we show that the symmetric decomposition for $\{a_{rn}\}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max \{s,d+1-s\}$. If the polytope is Gorenstein then this decomposition is moreover interlacing.