{
  "id": "2012.04299",
  "version": "v1",
  "published": "2020-12-08T09:18:59.000Z",
  "updated": "2020-12-08T09:18:59.000Z",
  "title": "Sign patterns and rigid moduli orders",
  "authors": [
    "Yousra Gati",
    "Vladimir Petrov Kostov",
    "Mohamed Chaouki Tarchi"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\\sum_{j=0}^{d-1}a_jx^j$ and its {\\em hyperbolicity domain} $\\Pi_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all roots real. The subset $E_d\\subset \\Pi_d$ is the one on which a modulus of a negative root of $Q_d$ is equal to a positive root of $Q_d$. At a point, where $Q_d$ has $d$ distinct roots with exactly $s$ ($1\\leq s\\leq [d/2]$) equalities between positive roots and moduli of negative roots, the set $E_d$ is locally the transversal intersection of $s$ smooth hypersurfaces. At a point, where $Q_d$ has two double opposite roots and no other equalities between moduli of roots, the set $E_d$ is locally the direct product of $\\mathbb{R}^{d-3}$ and a hypersurface in $\\mathbb{R}^3$ having a Whitney umbrella singularity. For $d\\leq 4$, we draw pictures of the sets $\\Pi_d$ and~$E_d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-08T09:18:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rigid moduli orders",
      "sign patterns",
      "real univariate polynomials",
      "negative root",
      "positive root"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}