{
  "id": "0812.3687",
  "version": "v3",
  "published": "2008-12-19T00:33:56.000Z",
  "updated": "2009-05-14T00:21:12.000Z",
  "title": "On multivariate Newton-like inequalities",
  "authors": [
    "Leonid Gurvits"
  ],
  "comment": "A publication version, a subsection on lower bounds on the inner product of H-Stable polynomials is added",
  "categories": [
    "math.CO",
    "math.OC"
  ],
  "abstract": "We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\\bf Strongly Log-Concave} entire functions, generalizing {\\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$ variables is called {\\bf Strongly Log-Concave} if the function $(\\partial x_1)^{c_1}...(\\partial x_m)^{c_m} f$ is either zero or $\\log((\\partial x_1)^{c_1}...(\\partial x_m)^{c_m} f)$ is concave on $R_{+}^{m}$. We start with yet another point of view (via {\\it propagation}) on the standard univarite (or homogeneous bivariate) {\\bf Newton Inequlities}. We prove analogues of (univariate) {\\bf Newton Inequlities} in the (multivariate) {\\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like) inequalities is the fact that the support $supp(f)$ of a {\\bf Strongly Log-Concave} entire function $f$ is discretely convex ($D$-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives $Der_{f}(r_1,...,r_m)$ of $f$ at zero and on the lower bounds on $Der_{f}(r_1,...,r_m)$, which generalize the {\\bf Van Der Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-05-14T00:21:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inequality",
      "multivariate newton-like inequalities",
      "strongly log-concave",
      "newton inequlities",
      "study multivariate entire functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.3687G"
    }
  }
}