{
  "id": "1806.08165",
  "version": "v1",
  "published": "2018-06-21T10:51:41.000Z",
  "updated": "2018-06-21T10:51:41.000Z",
  "title": "Interlacing Polynomials and the Veronese Construction for Rational Formal Power Series",
  "authors": [
    "Philip B. Zhang"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Fixing a positive integer $r$ and $0 \\le k \\le r-1$, define $f^{\\langle r,k \\rangle}$ for every formal power series $f$ as $ f(x) = f^{\\langle r,0 \\rangle}(x^r)+xf^{\\langle r,1 \\rangle}(x^r)+ \\cdots +x^{r-1}f^{\\langle r,r-1 \\rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\\, h(x) := \\left( (1+x+\\cdots+x^{r-1})^{n} h(x) \\right)^{\\langle r,k \\rangle}$ has only nonpositive zeros for any $r \\ge \\deg h(x) -k$ and any positive integer $n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which states that $U^{n}_{r,0}\\,h(x)$ has only negative, real zeros whenever $r\\ge n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $\\left( h^{\\langle r,r-i \\rangle}(x)\\right)_{1\\le i \\le r}$ is interlacing, so is $\\left( U^{n}_{r,r-i}\\, h(x) \\right)_{1\\le i \\le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-21T10:51:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "13A02",
      "26C10",
      "52B20",
      "52B45"
    ],
    "keywords": [
      "rational formal power series",
      "veronese construction",
      "interlacing polynomials",
      "positive integer",
      "colored permutations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}