{
  "id": "1811.04844",
  "version": "v1",
  "published": "2018-11-12T16:41:31.000Z",
  "updated": "2018-11-12T16:41:31.000Z",
  "title": "A Nonlocal Transport Equation Describing Roots of Polynomials Under Differentiation",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "Let $p_n$ be a polynomial of degree $n$ having all its roots on the real line distributed according to a smooth function $u(0,x)$. One could wonder how the distribution of roots behaves under iterated differentation of the function, i.e. how the density of roots of $p_n^{(k)}$ evolves. We derive a nonlinear transport equation with nonlocal flux $$ u_t + \\frac{1}{\\pi}\\left( \\arctan{ \\left( \\frac{Hu}{ u}\\right)} \\right)_x = 0,$$ where $H$ is the Hilbert transform. This equation has three very different compactly supported solutions: (1) the arcsine distribution $u(t,x) = (1-x^2)^{-1/2} \\chi_{(-1,1)}$, (2) the family of semicircle distributions $$ u(t,x) = \\frac{2}{\\pi} \\sqrt{(T-t) - x^2}$$ and (3) a family of solutions contained in the Marchenko-Pastur law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-12T16:41:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal transport equation",
      "polynomial",
      "differentiation",
      "nonlinear transport equation",
      "nonlocal flux"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}