{
  "id": "2310.14698",
  "version": "v1",
  "published": "2023-10-23T08:44:17.000Z",
  "updated": "2023-10-23T08:44:17.000Z",
  "title": "Degree $6$ hyperbolic polynomials and orders of moduli",
  "authors": [
    "Yousra Gati",
    "Vladimir Petrov Kostov",
    "Mohamed Chaouki Tarchi"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\\tilde{c}$ positive and $\\tilde{p}$ negative roots counted with multiplicity, where $\\tilde{c}$ and $\\tilde{p}$ are the numbers of sign changes and sign preservations in the sequence of its coefficients, $\\tilde{c}+\\tilde{p}=d$. For $d=6$, we give the exhaustive answer to the question: When the moduli of all $6$ roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-23T08:44:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "30C15"
    ],
    "keywords": [
      "hyperbolic polynomials",
      "real univariate degree",
      "negative roots",
      "sign changes",
      "signs implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}