{
  "id": "1812.04874",
  "version": "v1",
  "published": "2018-12-12T10:14:40.000Z",
  "updated": "2018-12-12T10:14:40.000Z",
  "title": "Exponential convexifying of polynomials",
  "authors": [
    "Krzysztof Kurdyka",
    "Katarzyna Kuta",
    "Stanisław Spodzieja"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AG",
    "math.CA"
  ],
  "abstract": "Let $X\\subset\\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\\geq 1$, such that for any $\\xi\\in\\mathbb{R}^n$ the function $\\varphi_N(x)=e^{N|x-\\xi|^2}f(x)$ is strongly convex on $X$. When $X$ is unbounded we have to assume also that the leading form of $f$ is positive in $\\mathbb{R}^n\\setminus\\{0\\}$. We obtain strong convexity of $\\varPhi_N(x)=e^{e^{N|x|^2}}f(x)$ on possibly unbounded $X$, provided $N$ is sufficiently large, assuming only that $f$ is positive on $X$. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-12T10:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "12D15",
      "26B25"
    ],
    "keywords": [
      "polynomial",
      "exponential convexifying",
      "convex closed semialgebraic sets",
      "strong convexity",
      "strongly convex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}