{
  "id": "2004.05423",
  "version": "v1",
  "published": "2020-04-11T15:13:11.000Z",
  "updated": "2020-04-11T15:13:11.000Z",
  "title": "Symmetric Decompositions and the Veronese Construction",
  "authors": [
    "Katharina Jochemko"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-11T15:13:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "13A02",
      "26C10",
      "52B20"
    ],
    "keywords": [
      "symmetric decomposition",
      "veronese construction",
      "numerator polynomial",
      "natural linear inequalities",
      "study rational generating functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}