{
  "id": "1802.02752",
  "version": "v1",
  "published": "2018-02-08T08:45:08.000Z",
  "updated": "2018-02-08T08:45:08.000Z",
  "title": "Degree bound of Pólya Positivstellenstaz",
  "authors": [
    "Ze Kang Tan"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "P\\'olya's Positivstellensatz on the $1$-simplex says that if $P(x)$ is a real polynomial such that $P(x)>0$ whenever $x \\ge 0$, then all the coefficients of $(1+x)^mP(x)$ are positive whenever $m$ is large. Powers-Reznick gave a complexity estimate for P\\'olya's Positivstellensatz. Namely, they proved that, for such $P(x)$ of degree $d$, all the coefficients of $(1+x)^mP(x)$ are positive whenever $m > \\frac{1}{2} (d^2 -d) \\frac{L(P)}{\\lambda(P)} - d$. where $\\frac{L(P)}{\\lambda(P)}$ is an invariant of $P(x)$. For $d=3$ and $d=4$ specifically, we improve Powers-Reznick's bound by showing $m > \\frac{3}{2} \\frac{L(P)}{\\lambda(P)} - 1$ for $d=3$ and $ m > \\frac{4232}{2505} \\frac{L(P)}{\\lambda(P)} - 1$ for $d=4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-08T08:45:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pólya positivstellenstaz",
      "degree bound",
      "polyas positivstellensatz",
      "powers-reznicks bound",
      "complexity estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}