{
  "id": "1701.07912",
  "version": "v1",
  "published": "2017-01-27T00:39:53.000Z",
  "updated": "2017-01-27T00:39:53.000Z",
  "title": "An extension of the Hermite-Biehler theorem with application to polynomials with one positive root",
  "authors": [
    "Richard Ellard",
    "Helena Šmigoc"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots interlace. This bound depends on the number of roots of $f$ which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive roots of a polynomial in terms of the number of sign changes in its coefficients. We use our extension of the Hermite-Biehler Theorem to give an inverse rule of signs for polynomials with one positive root.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-27T00:39:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "93D20",
      "15A29"
    ],
    "keywords": [
      "positive root",
      "application",
      "left half plane",
      "real roots interlace",
      "open left half-plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}