{
  "id": "1905.01836",
  "version": "v1",
  "published": "2019-05-06T06:34:28.000Z",
  "updated": "2019-05-06T06:34:28.000Z",
  "title": "On Descartes' rule of signs",
  "authors": [
    "Hassen Cheriha",
    "Yousra Gati",
    "Vladimir Petrov Kostov"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\\sum _{j=0}^{d-1}a_jx^j$, $a_j\\neq 0$, defines the SP $\\sigma :=(+$,sgn$(a_{d-1})$, $\\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of positive (resp. $neg$ of negative) roots of $P$, one has $pos\\leq c$ (resp. $neg\\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign preservations in $\\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say that $P$ realizes the SP $\\sigma$ with the pair $(pos, neg)$. For SPs with $c=2$, we give some sufficient conditions for the (non)realizability of pairs $(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\\ldots$, $[(d-2)/2]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-06T06:34:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient conditions",
      "real polynomial",
      "sign preservations",
      "sign changes",
      "realizability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}