{
  "id": "1402.6185",
  "version": "v3",
  "published": "2014-02-25T14:57:06.000Z",
  "updated": "2015-02-07T23:42:31.000Z",
  "title": "Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming",
  "authors": [
    "Sadik Iliman",
    "Timo de Wolff"
  ],
  "comment": "Revisions, a section about the constrained case was added, and an appendix was added; 21 pages, 1 figure",
  "categories": [
    "math.OC",
    "math.AG"
  ],
  "abstract": "In this article, we propose a geometric programming method in order to compute lower bounds for real polynomials. We provide new sufficient criteria for polynomials to be nonnegative as well as to have a sum of binomial squares representation. These criteria rely on the coefficients and the support of a polynomial. These generalize all previous criteria by Lasserre, Ghasemi, Marshall, Fidalgo and Kovacec to polynomials with arbitrary simplex Newton polytopes. This generalization yields a geometric programming approach for computing lower bounds for polynomials that significantly extends the geometric program proposed by Ghasemi and Marshall. Furthermore, it shows that geometric programming is strongly related to nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the authors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-26T10:51:32.000Z",
      "abstract": "In this article, we propose a geometric programming method in order to compute lower bounds for real polynomials. We provide a new sufficient criterion on the coefficients of a polynomial to be nonnegative as well as on the coefficients and the support to have a sum of binomial squares representation. It generalizes all previous criteria by Lasserre, Ghasemi, Marshall, Fidalgo and Kovacec from scaled standard simplex Newton polytopes to arbitrary simplex Newton polytopes. This generalization yields a geometric programming approach for computing lower bounds for polynomials that significantly extends the one proposed by Ghasemi and Marshall. Furthermore, it shows that geometric programming is strongly related to nonnegativity certificates given by sums of nonnegative circuit polynomials, which were recently introduced by the authors.",
      "comment": "Minor revisions and a significant improvement of the main result; 13 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-02-07T23:42:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12D15",
      "14P99",
      "52B20",
      "90C25"
    ],
    "keywords": [
      "lower bounds",
      "geometric programming",
      "arbitrary simplex newton polytopes",
      "scaled standard simplex newton polytopes",
      "binomial squares representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.6185I"
    }
  }
}